Research Article Stochastic Analysis of Gaussian Processes...

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Research Article Stochastic Analysis of Gaussian Processes via Fredholm Representation Tommi Sottinen 1 and Lauri Viitasaari 2 1 Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, 65101 Vaasa, Finland 2 Department of Mathematics and System Analysis, Aalto University School of Science, P.O. Box 11100, Helsinki, 00076 Aalto, Finland Correspondence should be addressed to Lauri Viitasaari; lauri.viitasaari@iki.fi Received 22 March 2016; Accepted 19 June 2016 Academic Editor: Ciprian A. Tudor Copyright © 2016 T. Sottinen and L. Viitasaari. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. We show the convenience of the Fredholm representation by giving applications to equivalence in law, bridges, series expansions, stochastic differential equations, and maximum likelihood estimations. 1. Introduction e stochastic analysis of Gaussian processes that are not semimartingales is challenging. One way to overcome the challenge is to represent the Gaussian process under con- sideration, , say, in terms of a Brownian motion, and then develop a transfer principle so that the stochastic analysis can be done in the “Brownian level” and then transferred back into the level of . One of the most studied representations in terms of a Brownian motion is the so-called Volterra representation. A Gaussian Volterra process is a process that can be represented as =∫ 0 (, ) d , ∈ [0, ] , (1) where is a Brownian motion and ∈ 2 ([0, ] 2 ). Here, the integration goes only up to , hence the name “Volterra.” is Volterra nature is very convenient: it means that the filtration of is included in the filtration of the underlying Brownian motion . Gaussian Volterra processes and their stochastic analysis have been studied, for example, in [1, 2], just to mention a few. Apparently, the most famous Gaussian process admitting Volterra representation is the fractional Brownian motion and its stochastic analysis indeed has been developed mostly by using its Volterra representation; see, for example, the monographs [3, 4] and references therein. In discrete finite time the Volterra representation (1) is nothing but the Cholesky lower-triangular factorization of the covariance of , and hence every Gaussian process is a Volterra process. In continuous time this is not true; see Example 16 in Section 3. ere is a more general representation than (1) by Hida; see [5, eorem 4.1]. However, this Hida representation includes possibly infinite number of Brownian motions. Consequently, it seems very difficult to apply the Hida rep- resentation to build a transfer principle needed by stochastic analysis. Moreover, the Hida representation is not quite general. Indeed, it requires, among other things, that the Gaussian process is purely nondeterministic. e Fredholm representation (2) does not require pure nondeterminism. Example 16 in Section 3, which admits a Fredholm represen- tation, does not admit a Hida representation, and the reason is the lack of pure nondeterminism. e problem with the Volterra representation (1) is the Volterra nature of the kernel , as far as generality is concerned. Indeed, if one considers Fredholm kernels, that is, kernels where the integration is over the entire interval [0, ] under consideration, one obtains generality. A Gaussian Hindawi Publishing Corporation International Journal of Stochastic Analysis Volume 2016, Article ID 8694365, 15 pages http://dx.doi.org/10.1155/2016/8694365

Transcript of Research Article Stochastic Analysis of Gaussian Processes...

Page 1: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

Research ArticleStochastic Analysis of Gaussian Processes via FredholmRepresentation

Tommi Sottinen1 and Lauri Viitasaari2

1Department of Mathematics and Statistics University of Vaasa PO Box 700 65101 Vaasa Finland2Department of Mathematics and System Analysis Aalto University School of Science PO Box 11100 Helsinki 00076 Aalto Finland

Correspondence should be addressed to Lauri Viitasaari lauriviitasaariikifi

Received 22 March 2016 Accepted 19 June 2016

Academic Editor Ciprian A Tudor

Copyright copy 2016 T Sottinen and L Viitasaari This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respectto a Brownian motion We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using itWe show the convenience of the Fredholm representation by giving applications to equivalence in law bridges series expansionsstochastic differential equations and maximum likelihood estimations

1 Introduction

The stochastic analysis of Gaussian processes that are notsemimartingales is challenging One way to overcome thechallenge is to represent the Gaussian process under con-sideration 119883 say in terms of a Brownian motion and thendevelop a transfer principle so that the stochastic analysis canbe done in the ldquoBrownian levelrdquo and then transferred backinto the level of119883

One of the most studied representations in terms of aBrownian motion is the so-called Volterra representation AGaussian Volterra process is a process that can be representedas

119883119905= int

119905

0

119870 (119905 119904) d119882119904 119905 isin [0 119879] (1)

where 119882 is a Brownian motion and 119870 isin 1198712([0 119879]2) Herethe integration goes only up to 119905 hence the name ldquoVolterrardquoThis Volterra nature is very convenient it means that thefiltration of 119883 is included in the filtration of the underlyingBrownian motion119882 Gaussian Volterra processes and theirstochastic analysis have been studied for example in [1 2]just to mention a few Apparently the most famous Gaussianprocess admitting Volterra representation is the fractional

Brownian motion and its stochastic analysis indeed has beendevelopedmostly by using its Volterra representation see forexample the monographs [3 4] and references therein

In discrete finite time the Volterra representation (1) isnothing but the Cholesky lower-triangular factorization ofthe covariance of 119883 and hence every Gaussian process isa Volterra process In continuous time this is not true seeExample 16 in Section 3

There is a more general representation than (1) by Hidasee [5 Theorem 41] However this Hida representationincludes possibly infinite number of Brownian motionsConsequently it seems very difficult to apply the Hida rep-resentation to build a transfer principle needed by stochasticanalysis Moreover the Hida representation is not quitegeneral Indeed it requires among other things that theGaussian process is purely nondeterministic The Fredholmrepresentation (2) does not require pure nondeterminismExample 16 in Section 3 which admits a Fredholm represen-tation does not admit a Hida representation and the reasonis the lack of pure nondeterminism

The problem with the Volterra representation (1) is theVolterra nature of the kernel 119870 as far as generality isconcerned Indeed if one considers Fredholm kernels that iskernels where the integration is over the entire interval [0 119879]under consideration one obtains generality A Gaussian

Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2016 Article ID 8694365 15 pageshttpdxdoiorg10115520168694365

2 International Journal of Stochastic Analysis

Fredholm process is a process that admits the Fredholmrepresentation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 119905 isin [0 119879] (2)

where119882 is a Brownian motion and 119870119879isin 119871

2([0 119879]

2) In this

paper we show that every separable Gaussian process withintegrable variance function admits representation (2) Theprice we have to pay for this generality is twofold

(i) The process 119883 is generated in principle from theentire path of the underlying Brownian motion 119882Consequently 119883 and 119882 do not necessarily generatethe same filtration This is unfortunate in manyapplications

(ii) In general the kernel 119870119879depends on 119879 even if the

covariance 119877 does not and consequently the derivedoperators also depend on 119879 This is why we use thecumbersome notation of explicitly stating out thedependence when there is one In stochastic analysisthis dependence on 119879 seems to be a minor inconve-nience however Indeed even in the Volterra case asexamined for example by Alos et al [1] one cannotavoid the dependence on 119879 in the transfer principleOf course for statistics where one would like to let 119879tend to infinity this is a major inconvenience

Let us note that the Fredholm representation has alreadybeen used without proof in [6] where the Holder continuityof Gaussian processes was studied

Let us mention a few papers that study stochastic anal-ysis of Gaussian processes here Indeed several differentapproaches have been proposed in the literature In partic-ular fractional Brownian motion has been a subject of activestudy (see the monographs [3 4] and references therein)More general Gaussian processes have been studied in thealready mentioned work by Alos et al [1] They consideredGaussian Volterra processes where the kernel satisfies certaintechnical conditions In particular their results cover frac-tional BrownianmotionwithHurst parameter119867 gt 14 LaterCheridito and Nualart [7] introduced an approach based onthe covariance function itself rather than on the Volterrakernel 119870 Kruk et al [8] developed stochastic calculus forprocesses having finite 2-planar variation especially coveringfractional Brownian motion 119867 ge 12 Moreover Kruk andRusso [9] extended the approach to cover singular covari-ances hence covering fractional Brownian motion119867 lt 12Furthermore Mocioalca and Viens [10] studied processeswhich are close to processeswith stationary incrementsMoreprecisely their results cover cases whereE(119883

119905minus119883

119904)2sim 120574

2(|119905minus

119904|) where 120574 satisfies some minimal regularity conditions Inparticular their results cover some processes which are noteven continuous Finally the latest development we are awareof is a paper by Lei and Nualart [11] who developed stochasticcalculus for processes having absolute continuous covarianceby using extended domain of the divergence introduced in[9] Finally we would like to mention Lebovits [12] whoused the S-transform approach and obtained similar results

to ours although his notion of integral is not elementary asours

The results presented in this paper give unified approachto stochastic calculus for Gaussian processes and only inte-grability of the variance function is required In particularour results cover processes that are not continuous

The paper is organized as followsSection 2 contains some preliminaries on Gaussian pro-

cesses and isonormal Gaussian processes and related Hilbertspaces

Section 3 provides the proof of the main theorem of thepaper the Fredholm representation

In Section 4 we extend the Fredholm representation toa transfer principle in three contexts of growing generalityFirst we prove the transfer principle for Wiener integralsin Section 41 then we use the transfer principle to definethe multiple Wiener integral in Section 42 and finally inSection 43 we prove the transfer principle for Malliavincalculus thus showing that the definition of multiple Wienerintegral via the transfer principle done in Section 42 isconsistent with the classical definitions involving Brownianmotion or other Gaussian martingales Indeed classicallyone defines the multiple Wiener integrals either by buildingan isometry with removed diagonals or by spanning higherchaos by using the Hermite polynomials In the generalGaussian case one cannot of course remove the diagonalsbut the Hermite polynomial approach is still valid We showthat this approach is equivalent to the transfer principle InSection 43 we also prove an Ito formula for general Gaussianprocesses and in Section 44 we extend the Ito formula evenfurther by using the technique of extended domain in thespirit of [7] This Ito formula is as far as we know themost general version for Gaussian processes existing in theliterature so far

Finally in Section 5 we show the power of the transferprinciple in some applications In Section 51 the transferprinciple is applied to the question of equivalence of lawof general Gaussian processes In Section 52 we show howone can construct net canonical-type representation forgeneralizedGaussian bridges that is for theGaussian processthat is conditioned by multiple linear functionals of its pathIn Section 53 the transfer principle is used to provide seriesexpansions for general Gaussian processes

2 Preliminaries

Our general setting is as follows let 119879 gt 0 be a fixed finitetime-horizon and let 119883 = (119883

119905)119905isin[0119879]

be a Gaussian processwith covariance119877 that may ormay not depend on119879 Withoutloss of any interesting generalitywe assume that119883 is centeredWe also make the very weak assumption that 119883 is separablein the sense of the following definition

Definition 1 (separability) The Gaussian process119883 is separa-ble if the Hilbert space 1198712(Ω 120590(119883)P) is separable

Example 2 If the covariance 119877 is continuous then 119883 isseparable In particular all continuousGaussian processes areseparable

International Journal of Stochastic Analysis 3

Definition 3 (associated operator) For a kernel Γ isin

1198712([0 119879]

2) one associates an operator on 1198712([0 119879]) also

denoted by Γ as

Γ119891 (119905) = int

119879

0

119891 (119904) Γ (119905 119904) d119904 (3)

Definition 4 (isonormal process) The isonormal processassociated with 119883 also denoted by 119883 is the Gaussian family(119883(ℎ) ℎ isin H

119879) where the Hilbert space H

119879= H

119879(119877) is

generated by the covariance 119877 as follows

(i) Indicators 1119905fl 1

[0119905) 119905 le 119879 belong toH

119879

(ii) H119879is endowed with the inner product ⟨1

119905 1

119904⟩H119879

fl119877(119905 119904)

Definition 4 states that 119883(ℎ) is the image of ℎ isin H119879in

the isometry that extends the relation

119883(1119905) fl 119883

119905(4)

linearly Consequently we can have the following definition

Definition 5 (Wiener integral) 119883(ℎ) is theWiener integral ofthe element ℎ isinH

119879with respect to119883 One will also denote

int

119879

0

ℎ (119905) d119883119905fl 119883 (ℎ) (5)

Remark 6 Eventually all the following will mean the same

119883(ℎ) = int

119879

0

ℎ (119905) d119883119905= int

119879

0

ℎ (119905) 120575119883119905= 119868

1198791(ℎ)

= 119868119879(ℎ)

(6)

Remark 7 TheHilbert spaceH119879is separable if and only if119883

is separable

Remark 8 Due to the completion under the inner product⟨sdot sdot⟩H119879

it may happen that the space H119879is not a space of

functions but contains distributions compare [13] for the caseof fractional Brownian motions with Hurst index bigger thanhalf

Definition 9 The function space H0

119879sub H

119879is the space

of functions that can be approximated by step-functions on[0 119879] in the inner product ⟨sdot sdot⟩H119879

Example 10 If the covariance 119877 is of bounded variation thenH0

119879is the space of functions 119891 satisfying

∬

119879

0

1003816100381610038161003816119891 (119905) 119891 (119904)1003816100381610038161003816 |119877| (d119904 d119905) lt infin (7)

Remark 11 Note that itmay be that119891 isinH0

119879but for some1198791015840 lt

119879 we have 11989111198791015840 notin H0

1198791015840 compare [14] for an example with

fractional Brownian motion with Hurst index less than halfFor this reason we keep the notation H

119879instead of simply

writing H For the same reason we include the dependenceof 119879 whenever there is one

3 Fredholm Representation

Theorem 12 (Fredholm representation) Let 119883 = (119883119905)119905isin[0119879]

be a separable centered Gaussian process Then there exists akernel119870

119879isin 119871

2([0 119879]

2) and a Brownian motion119882 = (119882

119905)119905ge0

independent of 119879 such that

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904(8)

in law if and only if the covariance 119877 of 119883 satisfies the tracecondition

int

119879

0

119877 (119905 119905) d119905 lt infin (9)

Representation (8) is unique in the sense that any otherrepresentation with kernel

119879 say is connected to (8) by

a unitary operator 119880 on 1198712([0 119879]) such that 119879= 119880119870

119879

Moreover one may assume that 119870119879is symmetric

Proof Let us first remark that (9) is precisely what we need toinvokeMercerrsquos theorem and take square root in the resultingexpansion

Now by Mercerrsquos theorem we can expand the covariancefunction 119877 on [0 119879]2 as

119877 (119905 119904) =

infin

sum

119894=1

120582119879

119894119890119879

119894(119905) 119890

119879

119894(119904) (10)

where (120582119879119894)infin

119894=1and (119890119879

119894)infin

119894=1are the eigenvalues and the eigen-

functions of the covariance operator

119877119879119891 (119905) = int

119879

0

119891 (119904) 119877 (119905 119904) d119904 (11)

Moreover (119890119879119894)infin

119894=1is an orthonormal system on 1198712([0 119879])

Now 119877119879 being a covariance operator admits a square

root operator119870119879defined by the relation

int

119879

0

119890119879

119894(119904) 119877

119879119890119879

119895(119904) d119904 = int

119879

0

119870119879119890119879

119894(119904) 119870

119879119890119879

119895(119904) d119904 (12)

for all 119890119879119894and 119890119879

119895 Now condition (9) means that 119877

119879is trace

class and consequently 119870119879is Hilbert-Schmidt operator In

particular 119870119879is a compact operator Therefore it admits

a kernel Indeed a kernel 119870119879can be defined by using the

Mercer expansion (10) as

119870119879(119905 119904) =

infin

sum

119894=1

radic120582119879

119894119890119879

119894(119905) 119890

119879

119894(119904) (13)

This kernel is obviously symmetric Now it follows that

119877 (119905 119904) = int

119879

0

119870119879(119905 119906)119870

119879(119904 119906) d119906 (14)

and representation (8) follows from thisFinally let us note that the uniqueness up to a unitary

transformation is obvious from the square root relation(12)

4 International Journal of Stochastic Analysis

Remark 13 The Fredholm representation (8) holds also forinfinite intervals that is 119879 = infin if the trace condition (9)holds Unfortunately this is seldom the case

Remark 14 The above proof shows that the Fredholm repre-sentation (8) holds in law However one can also constructthe process119883 via (8) for a given Brownian motion119882 In thiscase representation (8) holds of course in 1198712 Finally notethat in general it is not possible to construct the Brownianmotion in representation (8) from the process 119883 Indeedthere might not be enough randomness in 119883 To construct119882 from119883 one needs that the indicators 1

119905 119905 isin [0 119879] belong

to the range of the operator119870119879

Remark 15 We remark that the separability of 119883 ensuresrepresentation of form (8) where the kernel 119870

119879only satisfies

a weaker condition 119870119879(119905 sdot) isin 119871

2([0 119879]) for all 119905 isin [0 119879]

which may happen if the trace condition (9) fails In thiscase however the associated operator119870

119879does not belong to

1198712([0 119879]) which may be undesirable

Example 16 Let us consider the following very degeneratecase suppose 119883

119905= 119891(119905)120585 where 119891 is deterministic and 120585 is a

standard normal random variable Suppose 119879 gt 1 Then

119883119905= int

119879

0

119891 (119905) 1[01)(119904) d119882s (15)

So 119870119879(119905 119904) = 119891(119905)1

[01)(119904) Now if 119891 isin 1198712([0 119879]) then

condition (9) is satisfied and 119870119879isin 119871

2([0 119879]

2) On the other

hand even if 119891 notin 1198712([0 119879]) we can still write 119883 in form(15) However in this case the kernel 119870

119879does not belong to

1198712([0 119879]

2)

Example 17 Consider a truncated series expansion

119883119905=

119899

sum

119896=1

119890119879

119896(119905) 120585

119896 (16)

where 120585119896are independent standard normal random variables

and

119890119879

119896(119905) = int

119905

0

119879

119896(119904) d119904 (17)

where 119879119896 119896 isin N is an orthonormal basis in 1198712([0 119879]) Now

it is straightforward to check that this process is not purelynondeterministic (see [15] for definition) and consequently119883 cannot have Volterra representation while it is clear that119883 admits a Fredholm representation On the other hand bychoosing the functions 119879

119896to be the trigonometric basis on

1198712([0 119879])119883 is a finite-rank approximation of the Karhunen-

Loeve representation of standard Brownianmotion on [0 119879]Hence by letting 119899 tend to infinity we obtain the standardBrownian motion and hence a Volterra process

Example 18 Let119882 be a standard Brownian motion on [0 119879]and consider the Brownian bridge Now there are two rep-resentations of the Brownian bridge (see [16] and references

therein on the representations of Gaussian bridges) Theorthogonal representation is

119861119905= 119882

119905minus119905

119879119882

119879 (18)

Consequently 119861 has a Fredholm representation with kernel119870119879(119905 119904) = 1

119905(119904) minus 119905119879 The canonical representation of the

Brownian bridge is

119861t = (119879 minus 119905) int119905

0

1

119879 minus 119904d119882

119904 (19)

Consequently the Brownian bridge has also a Volterra-typerepresentation with kernel 119870(119905 119904) = (119879 minus 119905)(119879 minus 119904)

4 Transfer Principle and Stochastic Analysis

41 Wiener Integrals Theorem 22 is the transfer principle inthe context of Wiener integrals The same principle extendsto multiple Wiener integrals and Malliavin calculus later inthe following subsections

Recall that for any kernel Γ isin 1198712([0 119879]2) its associatedoperator on 1198712([0 119879]) is

Γ119891 (119905) = int

119879

0

119891 (119904) Γ (119905 119904) d119904 (20)

Definition 19 (adjoint associated operator) The adjoint asso-ciated operator Γlowast of a kernel Γ isin 1198712([0 119879]2) is defined bylinearly extending the relation

Γlowast1

119905= Γ (119905 sdot) (21)

Remark 20 Thename andnotation of ldquoadjointrdquo for119870lowast

119879comes

from Alos et al [1] where they showed that in their Volterracontext119870lowast

119879admits a kernel and is an adjoint of119870

119879in the sense

that

int

119879

0

119870lowast

119879119891 (119905) 119892 (119905) d119905 = int

119879

0

119891 (119905)119870119879119892 (d119905) (22)

for step-functions 119891 and 119892 belonging to 1198712([0 119879]) It isstraightforward to check that this statement is valid also inour case

Example 21 Suppose the kernel Γ(sdot 119904) is of bounded variationfor all 119904 and that 119891 is nice enough Then

Γlowast119891 (119904) = int

119879

0

119891 (119905) Γ (d119905 119904) (23)

Theorem 22 (transfer principle for Wiener integrals) Let 119883be a separable centered Gaussian process with representation(8) and let 119891 isinH

119879 Then

int

119879

0

119891 (119905) d119883119905= int

119879

0

119870lowast

119879119891 (119905) d119882

119905 (24)

Proof Assume first that 119891 is an elementary function of form

119891 (119905) =

119899

sum

119896=1

1198861198961119860119896

(25)

International Journal of Stochastic Analysis 5

for some disjoint intervals 119860119896= (119905

119896minus1 119905119896] Then the claim

follows by the very definition of the operator119870lowast

119879andWiener

integral with respect to 119883 together with representation(8) Furthermore this shows that 119870lowast

119879provides an isometry

between H119879and 1198712([0 119879]) Hence H

119879can be viewed as a

closure of elementary functions with respect to 119891H119879 =119870

lowast

119879119891

1198712([0119879])

which proves the claim

42 Multiple Wiener Integrals The study of multiple Wienerintegrals goes back to Ito [17] who studied the case ofBrownian motion Later Huang and Cambanis [18] extendedthe notion to general Gaussian processes Dasgupta andKallianpur [19 20] and Perez-Abreu and Tudor [21] studiedmultiple Wiener integrals in the context of fractional Brown-ian motion In [19 20] a method that involved a prior controlmeasure was used and in [21] a transfer principle was usedOur approach here extends the transfer principle methodused in [21]

We begin by recalling multiple Wiener integrals withrespect to Brownian motion and then we apply transferprinciple to generalize the theory to arbitrary Gaussianprocess

Let 119891 be an elementary function on [0 119879]119901 that vanisheson the diagonals that is

119891 =

119899

sum

1198941 119894119901=1

1198861198941 sdotsdotsdot119894119901

1Δ 1198941

timessdotsdotsdottimesΔ 119894119901 (26)

whereΔ119896fl [119905

119896minus1 119905119896) and 119886

1198941 119894119901= 0whenever 119894

119896= 119894

ℓfor some

119896 = ℓ For such 119891 we define the multiple Wiener integral as

119868119882

119879119901(119891) fl int

119879

0

sdot sdot sdot int

119879

0

119891 (1199051 119905

119901) 120575119882

1199051sdot sdot sdot 120575119882

119905119901

fl119899

sum

1198941i119901=11198861198941 119894119901Δ119882

1199051sdot sdot sdot Δ119882

119905119901

(27)

where we have denoted Δ119882119905119896fl119882

119905119896minus119882

119905119896minus1 For 119901 = 0 we set

119868119882

0(119891) = 119891 Now it can be shown that elementary functions

that vanish on the diagonals are dense in 1198712([0 119879]119901)Thusone can extend the operator 119868119882

119879119901to the space 1198712([0 119879]119901) This

extension is called the multiple Wiener integral with respectto the Brownian motion

Remark 23 It is well-known that 119868119882119879119901(119891) can be under-

stood as multiple or iterated Ito integrals if and only if119891(119905

1 119905

119901) = 0 unless 119905

1le sdot sdot sdot le 119905

119901 In this case we have

119868119882

119879119901(119891)

= 119901 int

119879

0

int

119905119901

0

sdot sdot sdot int

1199052

0

119891 (1199051 119905

119901) d119882

1199051sdot sdot sdot d119882

119905119901

(28)

For the case of Gaussian processes that are not martingalesthis fact is totally useless

For a general Gaussian process119883 recall first the Hermitepolynomials

119867119901(119909) fl

(minus1)119901

119901119890(12)119909

2 d119901

d119909119901(119890

minus(12)1199092

) (29)

For any 119901 ge 1 let the 119901th Wiener chaos of 119883 be theclosed linear subspace of 1198712(Ω) generated by the randomvariables 119867

119901(119883(120593)) 120593 isin H 120593H = 1 where 119867119901

is the119901th Hermite polynomial It is well-known that the mapping119868119883

119901(120593

otimes119901) = 119901119867

119901(119883(120593)) provides a linear isometry between

the symmetric tensor product H⊙119901 and the 119901th Wienerchaos The random variables 119868119883

119901(120593

otimes119901) are called multiple

Wiener integrals of order 119901 with respect to the Gaussianprocess119883

Let us now consider the multipleWiener integrals 119868119879119901

fora general Gaussian process119883 We define the multiple integral119868119879119901

by using the transfer principle in Definition 25 and laterargue that this is the ldquocorrectrdquo way of defining them So let119883 be a centered Gaussian process on [0 119879]with covariance 119877and representation (8) with kernel 119870

119879

Definition 24 (119901-fold adjoint associated operator) Let119870119879be

the kernel in (8) and let119870lowast

119879be its adjoint associated operator

Define

119870lowast

119879119901fl (119870lowast

119879)otimes119901

(30)

In the same way define

H119879119901

fl Hotimes119901

119879

H0

119879119901fl (Hotimes119901

119879)0

(31)

Here the tensor products are understood in the sense ofHilbert spaces that is they are closed under the inner productcorresponding to the 119901-fold product of the underlying innerproduct

Definition 25 Let 119883 be a centered Gaussian process withrepresentation (8) and let 119891 isinH

119879119901 Then

119868119879119901(119891) fl 119868119882

119879119901(119870

lowast

119879119901119891) (32)

The following example should convince the reader thatthis is indeed the correct definition

Example 26 Let 119901 = 2 and let ℎ = ℎ1otimesℎ

2 where both ℎ

1and

ℎ2are step-functions Then

(119870lowast

1198792ℎ) (119909 119910) = (119870

lowast

119879ℎ1) (119909) (119870

lowast

119879ℎ2) (119910)

119868119882

1198792(119870

lowast

1198792119891)

= int

119879

0

119870lowast

119879ℎ1(V) d119882V

sdot int

119879

0

119870lowast

119879ℎ2(119906) d119882

119906minus ⟨119870

lowast

119879ℎ1 119870

lowast

119879ℎ2⟩1198712([0119879])

= 119883 (ℎ1)119883 (ℎ

2) minus ⟨ℎ

1 ℎ

2⟩H119879

(33)

6 International Journal of Stochastic Analysis

The following proposition shows that our approach todefine multiple Wiener integrals is consistent with the tra-ditional approach where multiple Wiener integrals for moregeneral Gaussian process 119883 are defined as the closed linearspace generated by Hermite polynomials

Proposition 27 Let 119867119901be the 119901th Hermite polynomial and

let ℎ isinH119879 Then

119868119879119901(ℎ

otimes119901) = 119901 ℎ

119901

H119879119867119901(119883 (ℎ)

ℎH119879

) (34)

Proof First note that without loss of generalitywe can assumeℎH119879

= 1Now by the definition of multipleWiener integralwith respect to119883 we have

119868119879119901(ℎ

otimes119901) = 119868

119882

119879119901(119870

lowast

119879119901ℎotimes119901) (35)

where

119870lowast

119879119901ℎotimes119901= (119870

lowast

119879ℎ)

otimes119901

(36)

Consequently by [22 Proposition 114] we obtain

119868119879119901(ℎ

otimes119901) = 119901119867

119901(119882 (119870

lowast

119879ℎ)) (37)

which implies the result together withTheorem 22

Proposition 27 extends to the following product formulawhich is also well-known in the Gaussianmartingale case butapparently new for general Gaussian processes Again theproof is straightforward application of transfer principle

Proposition 28 Let 119891 isinH119879119901

and 119892 isinH119879119902Then

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimes

119870119879119903119892)

(38)

where

119891otimes119870119879119903119892 = (119870

lowast

119879119901+119902minus2119903)minus1

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (39)

and (119870lowast

119879119901+119902minus2119903)minus1 denotes the preimage of 119870lowast

119879119901+119902minus2119903

Proof The proof follows directly from the definition of119868119879119901(119891) and [22 Proposition 113]

Example 29 Let 119891 isin H119879119901

and 119892 isin H119879119902

be of forms119891(119909

1 119909

119901) = prod

119901

119896=1119891119896(119909

119896) and119892(119910

1 119910

119901) = prod

119902

119896=1119892119896(119910

119896)

Then

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (119909

1 119909

119901minus119903 119910

1 119910

119902minus119903)

= int[0119879]119903

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119903

prod

119895=1

119870lowast

119879119891119901minus119903+119895

(119904119895)

sdot

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot

119903

prod

119895=1

119870lowast

119879119892119902minus119903+119895(119904

119895) d119904

1sdot sdot sdot d119904

119903

=

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot ⟨

119903

prod

119895=1

119891119901minus119903+119895

119903

prod

119895=1

119892119902minus119903+119895⟩

H119879119903

(40)

Hence119891otimes

119870119879119903119892

=

119901minus119903

prod

119896=1

119891119896(119909

119896)

119902minus119903

prod

119896=1

119892119896(119910

119896)⟨

119903

prod

119895=1

119891119895

119903

prod

119895=1

119892119895⟩

H119879119903

(41)

Remark 30 In the literature multiple Wiener integrals areusually defined as the closed linear space spanned byHermitepolynomials In such a case Proposition 27 is clearly true bythe very definition Furthermore one has a multiplicationformula (see eg [23])

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimesH119879119903

119892)

(42)

where 119891otimesH119879119903119892 denotes symmetrization of tensor product

119891otimesH119879119903119892 =

infin

sum

1198941 119894119903=1

⟨119891 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

otimes ⟨119892 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

(43)

and 119890119896 119896 = 1 is a complete orthonormal basis of the

Hilbert space H119879 Clearly by Proposition 27 both formulas

coincide This also shows that (39) is well-defined

43 Malliavin Calculus and Skorohod Integrals We begin byrecalling some basic facts on Malliavin calculus

Definition 31 Denote by S the space of all smooth randomvariables of the form

119865 = 119891 (119883 (1205931) 119883 (120593

119899)) 120593

1 120593

119899isinH

119879 (44)

International Journal of Stochastic Analysis 7

where 119891 isin 119862infin119887(R119899) that is 119891 and all its derivatives are

bounded The Malliavin derivative 119863119879119865 = 119863

119883

119879119865 of 119865 is an

element of 1198712(ΩH119879) defined by

119863119879119865 =

119899

sum

119894=1

120597119894119891 (119883 (120593

1) 119883 (120593

119899)) 120593

119894 (45)

In particular119863119879119883119905= 1

119905

Definition 32 Let D12= D12

119883be the Hilbert space of all

square integrable Malliavin differentiable random variablesdefined as the closure of S with respect to norm

1198652

12= E [|119865|

2] + E [

10038171003817100381710038171198631198791198651003817100381710038171003817

2

H119879] (46)

The divergence operator 120575119879is defined as the adjoint

operator of the Malliavin derivative119863119879

Definition 33 The domain Dom 120575119879of the operator 120575

119879is the

set of random variables 119906 isin 1198712(ΩH119879) satisfying

10038161003816100381610038161003816E ⟨119863

119879119865 119906⟩

H119879

10038161003816100381610038161003816le 119888

119906119865

1198712 (47)

for any 119865 isin D12 and some constant 119888119906depending only on 119906

For 119906 isin Dom 120575119879the divergence operator 120575

119879(119906) is a square

integrable random variable defined by the duality relation

E [119865120575119879(119906)] = E ⟨119863

119879119865 119906⟩

H119879(48)

for all 119865 isin D12

Remark 34 It is well-known that D12sub Dom 120575

119879

We use the notation

120575119879(119906) = int

119879

0

119906119904120575119883

119904 (49)

Theorem 35 (transfer principle for Malliavin calculus) Let119883 be a separable centered Gaussian process with Fredholmrepresentation (8) Let 119863

119879and 120575

119879be the Malliavin derivative

and the Skorohod integral with respect to119883 on [0 119879] Similarlylet 119863119882

119879and 120575119882

119879be the Malliavin derivative and the Skorohod

integral with respect to the Brownianmotion119882 of (8) restrictedon [0 119879] Then

120575119879= 120575

119882

119879119870lowast

119879

119870lowast

119879119863119879= 119863

119882

119879

(50)

Proof The proof follows directly from transfer principle andthe isometry provided by 119870lowast

119879with the same arguments as in

[1] Indeed by isometry we have

H119879= (119870

lowast

119879)minus1

(1198712([0 119879])) (51)

where (119870lowast

119879)minus1 denotes the preimage which implies that

D12(H

119879) = (119870

lowast

119879)minus1

(D12

119882(119871

2([0 119879]))) (52)

which justifies 119870lowast

119879119863119879= 119863

119882

119879 Furthermore we have relation

E ⟨119906119863119879119865⟩

H119879= E ⟨119870

lowast

119879119906119863

119882

119879119865⟩

1198712([0119879])

(53)

for any smooth random variable 119865 and 119906 isin 1198712(ΩH119879)

Hence by the very definition ofDom 120575 and transfer principlewe obtain

Dom 120575119879= (119870

lowast

119879)minus1

(Dom 120575119882119879) (54)

and 120575119879(119906) = 120575

119882

119879(119870

lowast

119879119906) proving the claim

Now we are ready to show that the definition of themultiple Wiener integral 119868

119879119901in Section 42 is correct in the

sense that it agrees with the iterated Skorohod integral

Proposition 36 Let ℎ isin H119879119901

be of form ℎ(1199091 119909

119901) =

prod119901

119896=1ℎ119896(119909

119896) Then ℎ is iteratively 119901 times Skorohod integrable

and

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901) 120575119883

1199051sdot sdot sdot 120575119883

119905119901= 119868

119879119901(ℎ) (55)

Moreover if ℎ isin H0

119879119901is such that it is 119901 times iteratively

Skorohod integrable then (55) still holds

Proof Again the idea is to use the transfer principle togetherwith induction Note first that the statement is true for 119901 = 1by definition and assume next that the statement is validfor 119896 = 1 119901 We denote 119891

119895= prod

119895

119896=1ℎ119896(119909

119896) Hence by

induction assumption we have

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901 119905V) 1205751198831199051

sdot sdot sdot 120575119883119905119901120575119883V

= int

119879

0

119868119879119901(119891

119901) ℎ

119901+1(V) 120575119883V

(56)

Put now 119865 = 119868119879119901(119891

119901) and 119906(119905) = ℎ

119901+1(119905) Hence by [22

Proposition 133] and by applying the transfer principle weobtain that 119865119906 belongs to Dom 120575

119879and

120575119879(119865119906) = 120575

119879(119906) 119865 minus ⟨119863

119905119865 119906 (119905)⟩

H119879

= 119868119882

119879(119870

lowast

119879ℎ119901+1) 119868

119882

119879119901(119870

lowast

119879119901119891119901)

minus 119901 ⟨119868119879119901minus1(119891

119901(sdot 119905)) ℎ

119901+1(119905)⟩

H119879

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119882

119879119901minus1(119870

lowast

119879119901minus1119891119901otimes1119870lowast

119879ℎ119901+1)

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119879119901minus1(119891

119901otimes1198701198791ℎ119901+1)

(57)

Hence the result is valid also for 119901+ 1 by Proposition 28 with119902 = 1

The claim for general ℎ isinH0

119879119901follows by approximating

with a product of simple function

8 International Journal of Stochastic Analysis

Remark 37 Note that (55) does not hold for arbitrary ℎ isinH0

119879119901in general without the a priori assumption of 119901 times

iterative Skorohod integrability For example let 119901 = 2 and119883 = 119861

119867 be a fractional Brownian motion with119867 le 14 anddefine ℎ

119905(119904 V) = 1

119905(119904)1

119904(V) for some fixed 119905 isin [0 119879] Then

int

119879

0

ℎ119905(119904 V) 120575119883V = 119883119904

1119905(119904) (58)

But119883sdot1119905does not belong to Dom 120575

119879(see [7])

We end this section by providing an extension of Ito for-mulas provided by Alos et al [1] They considered GaussianVolterra processes that is they assumed the representation

119883119905= int

119905

0

119870 (119905 119904) d119882119904 (59)

where the kernel119870 satisfied certain technical assumptions In[1] it was proved that in the case of Volterra processes one has

119891 (119883119905) = 119891 (0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(60)

if 119891 satisfies the growth condition

max [1003816100381610038161003816119891 (119909)1003816100381610038161003816 100381610038161003816100381610038161198911015840(119909)100381610038161003816100381610038161003816100381610038161003816100381611989110158401015840(119909)10038161003816100381610038161003816] le 119888119890

120582|119909|2

(61)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1 In the

following we will consider different approach which enablesus to

(i) prove that such formula holds with minimal require-ments

(ii) give more instructive proof of such result(iii) extend the result from Volterra context to more

general Gaussian processes(iv) drop some technical assumptions posed in [1]

For simplicity we assume that the variance of 119883 is ofbounded variation to guarantee the existence of the integral

int

119879

0

11989110158401015840(119883

119905) d119877 (119905 119905) (62)

If the variance is not of bounded variation then integral (62)may be understood by integration by parts if 11989110158401015840 is smoothenough or in the general case via the inner product ⟨sdot sdot⟩H119879 InTheorem 40 we also have to assume that the variance of119883is bounded

The result for polynomials is straightforward once weassume that the paths of polynomials of 119883 belong to1198712(ΩH

119879)

Proposition 38 (Ito formula for polynomials) Let 119883 be aseparable centered Gaussian process with covariance 119877 andassume that 119901 is a polynomial Furthermore assume that for

each polynomial 119901 one has 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) Then for

each 119905 isin [0 119879] one has

119901 (119883119905) = 119901 (119883

0) + int

119905

0

1199011015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11990110158401015840(119883

119904) d119877 (119904 119904)

(63)

if and only if119883sdot1119905belongs to Dom 120575

119879

Remark 39 The message of the above result is that once theprocesses 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) then they automatically

belong to the domain of 120575119879which is a subspace of 1198712(ΩH

119879)

However in order to check 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) one needs

more information on the kernel 119870119879 A sufficient condition is

provided in Corollary 43 which covers many cases of interest

Proof By definition and applying transfer principle we haveto prove that 1199011015840(119883

sdot)1

119905belongs to domain of 120575

119879and that

Eint119905

0

119863119904119866119870

lowast

119879[119901

1015840(119883

sdot) 1

119905] d119904

= E [119866119901 (119883119905)] minus E [119866119901 (119883

0)]

minus1

2int

119905

0

E [11986611990110158401015840(119883

119905)] d119877 (119905 119905)

(64)

for every random variable 119866 from a total subset of 1198712(Ω)In other words it is sufficient to show that (69) is valid forrandom variables of form 119866 = 119868119882

119899(ℎ

otimes119899) where ℎ is a step-

functionNote first that it is sufficient to prove the claim only for

Hermite polynomials119867119896 119896 = 1 Indeed it is well-known

that any polynomial can be expressed as a linear combinationof Hermite polynomials and consequently the result forarbitrary polynomial 119901 follows by linearity

We proceed by induction First it is clear that firsttwo polynomials 119867

0and 119867

1satisfy (64) Furthermore by

assumption 1198671015840

2(119883

sdot)1

119905belongs to Dom 120575

119879from which (64)

is easily deduced by [22 Proposition 133] Assume nextthat the result is valid for Hermite polynomials 119867

119896 119896 =

0 1 119899 Then recall well-known recursion formulas

119867119899+1(119909) = 119909119867

119899(119909) minus 119899119867

119899minus1(119909)

1198671015840

119899(119909) = 119899119867

119899minus1(119909)

(65)

The induction step follows with straightforward calcula-tions by using the recursion formulas above and [22Proposition 133] We leave the details to the reader

We will now illustrate how the result can be generalizedfor functions satisfying the growth condition (61) by usingProposition 38 First note that the growth condition (61) isindeed natural since it guarantees that the left side of (60)is square integrable Consequently since operator 120575

119879is a

mapping from 1198712(ΩH119879) into 1198712(Ω) functions satisfying

(61) are the largest class of functions for which (60) can holdHowever it is not clear in general whether 1198911015840(119883

sdot)1

119905belongs

International Journal of Stochastic Analysis 9

to Dom 120575119879 Indeed for example in [1] the authors posed

additional conditions on the Volterra kernel 119870 to guaranteethis As our main result we show that E1198911015840(119883

sdot)1

1199052

H119879lt infin

implies that (60) holds In other words not only is the Itoformula (60) natural but it is also the only possibility

Theorem 40 (Ito formula for Skorohod integrals) Let 119883 bea separable centered Gaussian process with covariance 119877 suchthat all the polynomials 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) Assume that

119891 isin 1198622 satisfies growth condition (61) and that the variance of

119883 is bounded and of bounded variation If

E100381710038171003817100381710038171198911015840(119883

sdot) 1

119905

10038171003817100381710038171003817

2

H119879lt infin (66)

for any 119905 isin [0 119879] then

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(67)

Proof In this proof we assume for notational simplicity andwith no loss of generality that sup

0le119904le119879119877(119904 119904) = 1

First it is clear that (66) implies that 1198911015840(119883sdot)1

119905belongs to

domain of 120575119879 Hence we only have to prove that

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E [119866119891 (119883119905)] minus E [119866119891 (119883

0)]

minus1

2int

119905

0

E [11986611989110158401015840(119883

119904)] d119877 (119904 119904)

(68)

for every random variable 119866 = 119868119882119899(ℎ

otimes119899)

Now it is well-known that Hermite polynomials whenproperly scaled form an orthogonal system in 1198712(R) whenequipped with the Gaussian measure Now each 119891 satisfyingthe growth condition (61) has a series representation

119891 (119909) =

infin

sum

119896=0

120572119896119867119896(119909) (69)

Indeed the growth condition (61) implies that

intR

100381610038161003816100381610038161198911015840(119909)10038161003816100381610038161003816

2

119890minus1199092(2sup

0le119904le119879119877(119904119904))d119909 lt infin (70)

Furthermore we have

119891 (119883119904) =

infin

sum

119896=0

120572119896119867119896(119883

119904) (71)

where the series converge almost surely and in 1198712(Ω) andsimilar conclusion is valid for derivatives1198911015840(119883

119904) and 11989110158401015840(119883

119904)

Then by applying (66) we obtain that for any 120598 gt 0 thereexists119873 = 119873

120598such that we have

E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598 119899 ge 119873 (72)

where

1198911015840

119899(119883

119904) =

infin

sum

119896=119899

120572119896119867

1015840

119896(119883

119904) (73)

Consequently for random variables of form 119866 = 119868119882119899(ℎ

otimes119899) we

obtain by choosing 119873 large enough and applying Proposi-tion 38 that

E [119866119891 (119883119905)] minus E [119866119891 (119883

0)] minus1

2

sdot int

119905

0

E (11986611989110158401015840(119883

119905)) d119877 (119905 119905)

minus E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598

(74)

Now the left side does not depend on 119899 which concludes theproof

Remark 41 Note that actually it is sufficient to have

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

=

infin

sum

119896=1

120572119896E ⟨119863

119879119866119867

1015840

119896(119883

sdot) 1

119905⟩H119879

(75)

from which the result follows by Proposition 38 Further-more taking account growth condition (61) this is actuallysufficient and necessary condition for formula (60) to holdConsequently our method can also be used to obtain Itoformulas by considering extended domain of 120575

119879(see [9] or

[11]) This is the topic of Section 44

Example 42 It is known that if 119883 = 119861119867 is a fractional Brow-nian motion with119867 gt 14 then 1198911015840(119883

sdot)1

119905satisfies condition

(66) while for 119867 le 14 it does not (see [22 Chapter 5])Consequently a simple application of Theorem 40 coversfractional Brownian motion with 119867 gt 14 For the case119867 le 14 one has to consider extended domain of 120575

119879which is

proved in [9] Consequently in this case we have (75) for any119865 isin S

We end this section by illustrating the power of ourmethod with the following simple corollary which is anextension of [1 Theorem 1]

Corollary 43 Let 119883 be a separable centered continuousGaussian process with covariance 119877 that is bounded such thatthe Fredholm kernel 119870

119879is of bounded variation and

int

119879

0

(int

119879

0

1003817100381710038171003817119883119905minus 119883

119904

10038171003817100381710038171198712(Ω)

10038161003816100381610038161198701198791003816100381610038161003816 (d119905 119904))

2

d119904 lt infin (76)

Then for any 119905 isin [0 119879] one has

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(77)

10 International Journal of Stochastic Analysis

Proof Note that assumption is a Fredholm version of con-dition (K2) in [1] which implies condition (66) Hence theresult follows byTheorem 40

44 Extended Divergence Operator As shown in Section 43the Ito formula (60) is the only possibility However theproblem is that the space 1198712(ΩH

119879) may be too small to

contain the elements 1198911015840(119883sdot)1

119905 In particular it may happen

that not even the process 119883 itself belongs to 1198712(ΩH119879) (see

eg [7] for the case of fractional Brownian motion with119867 le 14) This problem can be overcome by consideringan extended domain of 120575

119879 The idea of extended domain is

to extend the inner product ⟨119906 120593⟩H119879 for simple 120593 to moregeneral processes 119906 and then define extended domain by(47) with a restricted class of test variables 119865 This also givesanother intuitive reason why extended domain of 120575

119879can be

useful indeed here we have proved that Ito formula (60) isthe only possibility and what one essentially needs for suchresult is the following

(i) 119883sdot1119905belongs to Dom 120575

119879

(ii) Equation (75) is valid for functions satisfying (61)

Consequently one should look for extensions of operator 120575119879

such that these two things are satisfiedTo facilitate the extension of domain we make the

following relatively moderate assumption

(H) The function 119905 997891rarr 119877(119905 119904) is of bounded variation on[0 119879] and

sup119905isin[0119879]

int

119879

0

|119877| (d119904 119905) lt infin (78)

Remark 44 Note that we are making the assumption on thecovariance 119877 not the kernel 119870

119879 Hence our case is different

from that of [1] Also [11] assumed absolute continuity in 119877we are satisfied with bounded variation

We will follow the idea from Lei and Nualart [11] andextend the inner product ⟨sdot sdot⟩H119879 beyondH

119879

Consider a step-function 120593Then on the one hand by theisometry property we have

⟨120593 1119905⟩H119879= int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 (79)

where 119892119905(119904) = 119870(119905 119904) isin 119871

2([0 119879]) On the other hand by

using adjoint property (see Remark 20) we obtain

int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 = int

119879

0

120593 (119904) (119870119879119892119905) (d119904) (80)

where computing formally we have

(119870119879119892119905) (d119904) = int

119879

0

119892119905(119906)119870

119879(d119904 119906) d119906

= int

119879

0

119870119879(119905 119906)119870

119879(d119904 119906) d119906 = 119877 (119905 d119904)

(81)

Consequently

⟨120593 1119905⟩H119879= int

119879

0

120593 (119904) 119877 (119905 d119904) (82)

This gives motivation to the following definition similar tothat of [11 Definition 21]

Definition 45 Denote by T119879the space of measurable func-

tions 119892 satisfying

sup119905isin[0119879]

int

119879

0

1003816100381610038161003816119892 (119904)1003816100381610038161003816 |119877| (119905 d119904) lt infin (83)

and let 120593 be a step-function of form 120593 = sum119899119896=11198871198961119905119896 Then we

extend ⟨sdot sdot⟩H119879 toT119879by defining

⟨119892 120593⟩H119879=

119899

sum

119896=1

119887119896int

119879

0

119892 (119904) 119877 (119905119896 d119904) (84)

In particular this implies that for 119892 and 120593 as above we have

⟨1198921119905 120593⟩

H119879= int

119905

0

119892 (119904) d ⟨1119904 120593⟩

H119879 (85)

We define extended domain Dom119864120575119879similarly as in [11]

Definition 46 A process 119906 isin 1198711(ΩT119879) belongs to Dom119864

120575119879

if10038161003816100381610038161003816E ⟨119906119863119865⟩H119879

10038161003816100381610038161003816le 119888

119906119865

2(86)

for any smooth random variable 119865 isin S In this case 120575(119906) isin1198712(Ω) is defined by duality relationship

E [119865120575 (119906)] = E ⟨119906119863119865⟩H119879 (87)

Remark 47 Note that in general Dom 120575119879and Dom119864

120575119879are

not comparable See [11] for discussion

Note now that if a function 119891 satisfies the growthcondition (61) then 1198911015840(119883

sdot)1

119905isin 119871

1(ΩT

119879) since (61) implies

E sup119905isin[0119879]

100381610038161003816100381610038161198911015840(119883

119905)10038161003816100381610038161003816

119901

lt infin (88)

for any 119901 lt (12120582)(sup119905isin[0119879]

119877(119905 119905))minus1 Consequently with

this definition we are able to get rid of the problem thatprocesses might not belong to corresponding H

119879-spaces

Furthermore this implies that the series expansion (69)converges in the norm 1198711(ΩT

119879) defined by

Eint119879

0

|119906 (119904)| |119877| (119905 d119904) (89)

which in turn implies (75) Hence it is straightforward toobtain the following by first showing the result for polynomi-als and then by approximating in a similar manner as done inSection 43 but using the extended domain instead

International Journal of Stochastic Analysis 11

Theorem 48 (Ito formula for extended Skorohod integrals)Let119883 be a separable centered Gaussian process with covariance119877 and assume that 119891 isin 1198622 satisfies growth condition (61)Furthermore assume that (H) holds and that the variance of119883is bounded and is of bounded variationThen for any 119905 isin [0 119879]the process 1198911015840(119883

sdot)1

119905belongs to Dom119864

120575119879and

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(90)

Remark 49 As an application of Theorem 48 it is straight-forward to derive version of Ito-Tanaka formula underadditional conditions which guarantee that for a certainsequence of functions 119891

119899we have the convergence of term

(12) int119905

011989110158401015840

119899(119883

119904)d119877(119904 119904) to the local time For details we refer

to [11] where authors derived such formula under theirassumptions

Finally let us note that the extension to functions 119891(119905 119909)is straightforward where 119891 satisfies the following growthcondition

max [1003816100381610038161003816119891 (119905 119909)1003816100381610038161003816 1003816100381610038161003816120597119905119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119909119891 (119905 119909)

1003816100381610038161003816]

le 119888119890120582|119909|2

(91)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1

Theorem 50 (Ito formula for extended Skorohod integralsII) Let 119883 be a separable centered Gaussian process withcovariance 119877 and assume that 119891 isin 11986212 satisfies growthcondition (91) Furthermore assume that (H) holds and thatthe variance of119883 is bounded and is of bounded variationThenfor any 119905 isin [0 119879] the process 120597

119909119891(sdot 119883

sdot)1

119905belongs to Dom119864

120575119879

and

119891 (119905 119883119905) = 119891 (0 119883

0) + int

119905

0

120597119909119891 (119904 119883

119904) 120575119883

119904

+ int

119905

0

120597119905119891 (119904 119883

119904) d119904

+1

2int

119905

0

120597119909119909119891 (119904 119883

119904) d119877 (119904 119904)

(92)

Proof Taking into account that we have no problems con-cerning processes to belong to the required spaces theformula follows by approximating with polynomials of form119901(119909)119902(119905) and following the proof of Theorem 40

5 Applications

We illustrate how some results transfer easily from theBrownian case to the Gaussian Fredholm processes

51 Equivalence in Law The transfer principle has alreadybeen used in connection with the equivalence of law of

Gaussian processes in for example [24] in the context offractional Brownian motions and in [2] in the context ofGaussianVolterra processes satisfying certain nondegeneracyconditions The following proposition uses the Fredholmrepresentation (8) to give a sufficient condition for theequivalence of general Gaussian processes in terms of theirFredholm kernels

Proposition 51 Let 119883 and be two Gaussian processes withFredholm kernels 119870

119879and

119879 respectively If there exists a

Volterra kernel ℓ isin 1198712([0 119879]2) such that

119879(119905 119904) = 119870

119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 (93)

then119883 and are equivalent in law

Proof Recall that by the Hitsuda representation theorem [5Theorem 63] a centered Gaussian process is equivalent inlaw to a Brownian motion on [0 119879] if and only if there existsa kernel ℓ isin 1198712([0 119879]2) and a Brownian motion119882 such that admits the representation

119905= 119882

119905minus int

119905

0

int

119904

0

ℓ (119904 119906) d119882119906d119904 (94)

Let119883 have the Fredholm representations

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (95)

Then is equivalent to119883 if it admits in law the representa-tion

119905

119889

= int

119879

0

119870119879(119905 119904) d

119904 (96)

where is connected to119882 of (95) by (94)In order to show (96) let

119905= int

119879

0

119879(119905 119904) d1198821015840

119905(97)

be the Fredholm representation of Here 1198821015840 is someBrownian motion Then by using connection (93) and theFubini theorem we obtain

119905= int

119879

0

119879(119905 119904) d1198821015840

119904

119889

= int

119879

0

119879(119905 119904) d119882

119904

= int

119879

0

(119870119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906) d119882

119904

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

2 International Journal of Stochastic Analysis

Fredholm process is a process that admits the Fredholmrepresentation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 119905 isin [0 119879] (2)

where119882 is a Brownian motion and 119870119879isin 119871

2([0 119879]

2) In this

paper we show that every separable Gaussian process withintegrable variance function admits representation (2) Theprice we have to pay for this generality is twofold

(i) The process 119883 is generated in principle from theentire path of the underlying Brownian motion 119882Consequently 119883 and 119882 do not necessarily generatethe same filtration This is unfortunate in manyapplications

(ii) In general the kernel 119870119879depends on 119879 even if the

covariance 119877 does not and consequently the derivedoperators also depend on 119879 This is why we use thecumbersome notation of explicitly stating out thedependence when there is one In stochastic analysisthis dependence on 119879 seems to be a minor inconve-nience however Indeed even in the Volterra case asexamined for example by Alos et al [1] one cannotavoid the dependence on 119879 in the transfer principleOf course for statistics where one would like to let 119879tend to infinity this is a major inconvenience

Let us note that the Fredholm representation has alreadybeen used without proof in [6] where the Holder continuityof Gaussian processes was studied

Let us mention a few papers that study stochastic anal-ysis of Gaussian processes here Indeed several differentapproaches have been proposed in the literature In partic-ular fractional Brownian motion has been a subject of activestudy (see the monographs [3 4] and references therein)More general Gaussian processes have been studied in thealready mentioned work by Alos et al [1] They consideredGaussian Volterra processes where the kernel satisfies certaintechnical conditions In particular their results cover frac-tional BrownianmotionwithHurst parameter119867 gt 14 LaterCheridito and Nualart [7] introduced an approach based onthe covariance function itself rather than on the Volterrakernel 119870 Kruk et al [8] developed stochastic calculus forprocesses having finite 2-planar variation especially coveringfractional Brownian motion 119867 ge 12 Moreover Kruk andRusso [9] extended the approach to cover singular covari-ances hence covering fractional Brownian motion119867 lt 12Furthermore Mocioalca and Viens [10] studied processeswhich are close to processeswith stationary incrementsMoreprecisely their results cover cases whereE(119883

119905minus119883

119904)2sim 120574

2(|119905minus

119904|) where 120574 satisfies some minimal regularity conditions Inparticular their results cover some processes which are noteven continuous Finally the latest development we are awareof is a paper by Lei and Nualart [11] who developed stochasticcalculus for processes having absolute continuous covarianceby using extended domain of the divergence introduced in[9] Finally we would like to mention Lebovits [12] whoused the S-transform approach and obtained similar results

to ours although his notion of integral is not elementary asours

The results presented in this paper give unified approachto stochastic calculus for Gaussian processes and only inte-grability of the variance function is required In particularour results cover processes that are not continuous

The paper is organized as followsSection 2 contains some preliminaries on Gaussian pro-

cesses and isonormal Gaussian processes and related Hilbertspaces

Section 3 provides the proof of the main theorem of thepaper the Fredholm representation

In Section 4 we extend the Fredholm representation toa transfer principle in three contexts of growing generalityFirst we prove the transfer principle for Wiener integralsin Section 41 then we use the transfer principle to definethe multiple Wiener integral in Section 42 and finally inSection 43 we prove the transfer principle for Malliavincalculus thus showing that the definition of multiple Wienerintegral via the transfer principle done in Section 42 isconsistent with the classical definitions involving Brownianmotion or other Gaussian martingales Indeed classicallyone defines the multiple Wiener integrals either by buildingan isometry with removed diagonals or by spanning higherchaos by using the Hermite polynomials In the generalGaussian case one cannot of course remove the diagonalsbut the Hermite polynomial approach is still valid We showthat this approach is equivalent to the transfer principle InSection 43 we also prove an Ito formula for general Gaussianprocesses and in Section 44 we extend the Ito formula evenfurther by using the technique of extended domain in thespirit of [7] This Ito formula is as far as we know themost general version for Gaussian processes existing in theliterature so far

Finally in Section 5 we show the power of the transferprinciple in some applications In Section 51 the transferprinciple is applied to the question of equivalence of lawof general Gaussian processes In Section 52 we show howone can construct net canonical-type representation forgeneralizedGaussian bridges that is for theGaussian processthat is conditioned by multiple linear functionals of its pathIn Section 53 the transfer principle is used to provide seriesexpansions for general Gaussian processes

2 Preliminaries

Our general setting is as follows let 119879 gt 0 be a fixed finitetime-horizon and let 119883 = (119883

119905)119905isin[0119879]

be a Gaussian processwith covariance119877 that may ormay not depend on119879 Withoutloss of any interesting generalitywe assume that119883 is centeredWe also make the very weak assumption that 119883 is separablein the sense of the following definition

Definition 1 (separability) The Gaussian process119883 is separa-ble if the Hilbert space 1198712(Ω 120590(119883)P) is separable

Example 2 If the covariance 119877 is continuous then 119883 isseparable In particular all continuousGaussian processes areseparable

International Journal of Stochastic Analysis 3

Definition 3 (associated operator) For a kernel Γ isin

1198712([0 119879]

2) one associates an operator on 1198712([0 119879]) also

denoted by Γ as

Γ119891 (119905) = int

119879

0

119891 (119904) Γ (119905 119904) d119904 (3)

Definition 4 (isonormal process) The isonormal processassociated with 119883 also denoted by 119883 is the Gaussian family(119883(ℎ) ℎ isin H

119879) where the Hilbert space H

119879= H

119879(119877) is

generated by the covariance 119877 as follows

(i) Indicators 1119905fl 1

[0119905) 119905 le 119879 belong toH

119879

(ii) H119879is endowed with the inner product ⟨1

119905 1

119904⟩H119879

fl119877(119905 119904)

Definition 4 states that 119883(ℎ) is the image of ℎ isin H119879in

the isometry that extends the relation

119883(1119905) fl 119883

119905(4)

linearly Consequently we can have the following definition

Definition 5 (Wiener integral) 119883(ℎ) is theWiener integral ofthe element ℎ isinH

119879with respect to119883 One will also denote

int

119879

0

ℎ (119905) d119883119905fl 119883 (ℎ) (5)

Remark 6 Eventually all the following will mean the same

119883(ℎ) = int

119879

0

ℎ (119905) d119883119905= int

119879

0

ℎ (119905) 120575119883119905= 119868

1198791(ℎ)

= 119868119879(ℎ)

(6)

Remark 7 TheHilbert spaceH119879is separable if and only if119883

is separable

Remark 8 Due to the completion under the inner product⟨sdot sdot⟩H119879

it may happen that the space H119879is not a space of

functions but contains distributions compare [13] for the caseof fractional Brownian motions with Hurst index bigger thanhalf

Definition 9 The function space H0

119879sub H

119879is the space

of functions that can be approximated by step-functions on[0 119879] in the inner product ⟨sdot sdot⟩H119879

Example 10 If the covariance 119877 is of bounded variation thenH0

119879is the space of functions 119891 satisfying

∬

119879

0

1003816100381610038161003816119891 (119905) 119891 (119904)1003816100381610038161003816 |119877| (d119904 d119905) lt infin (7)

Remark 11 Note that itmay be that119891 isinH0

119879but for some1198791015840 lt

119879 we have 11989111198791015840 notin H0

1198791015840 compare [14] for an example with

fractional Brownian motion with Hurst index less than halfFor this reason we keep the notation H

119879instead of simply

writing H For the same reason we include the dependenceof 119879 whenever there is one

3 Fredholm Representation

Theorem 12 (Fredholm representation) Let 119883 = (119883119905)119905isin[0119879]

be a separable centered Gaussian process Then there exists akernel119870

119879isin 119871

2([0 119879]

2) and a Brownian motion119882 = (119882

119905)119905ge0

independent of 119879 such that

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904(8)

in law if and only if the covariance 119877 of 119883 satisfies the tracecondition

int

119879

0

119877 (119905 119905) d119905 lt infin (9)

Representation (8) is unique in the sense that any otherrepresentation with kernel

119879 say is connected to (8) by

a unitary operator 119880 on 1198712([0 119879]) such that 119879= 119880119870

119879

Moreover one may assume that 119870119879is symmetric

Proof Let us first remark that (9) is precisely what we need toinvokeMercerrsquos theorem and take square root in the resultingexpansion

Now by Mercerrsquos theorem we can expand the covariancefunction 119877 on [0 119879]2 as

119877 (119905 119904) =

infin

sum

119894=1

120582119879

119894119890119879

119894(119905) 119890

119879

119894(119904) (10)

where (120582119879119894)infin

119894=1and (119890119879

119894)infin

119894=1are the eigenvalues and the eigen-

functions of the covariance operator

119877119879119891 (119905) = int

119879

0

119891 (119904) 119877 (119905 119904) d119904 (11)

Moreover (119890119879119894)infin

119894=1is an orthonormal system on 1198712([0 119879])

Now 119877119879 being a covariance operator admits a square

root operator119870119879defined by the relation

int

119879

0

119890119879

119894(119904) 119877

119879119890119879

119895(119904) d119904 = int

119879

0

119870119879119890119879

119894(119904) 119870

119879119890119879

119895(119904) d119904 (12)

for all 119890119879119894and 119890119879

119895 Now condition (9) means that 119877

119879is trace

class and consequently 119870119879is Hilbert-Schmidt operator In

particular 119870119879is a compact operator Therefore it admits

a kernel Indeed a kernel 119870119879can be defined by using the

Mercer expansion (10) as

119870119879(119905 119904) =

infin

sum

119894=1

radic120582119879

119894119890119879

119894(119905) 119890

119879

119894(119904) (13)

This kernel is obviously symmetric Now it follows that

119877 (119905 119904) = int

119879

0

119870119879(119905 119906)119870

119879(119904 119906) d119906 (14)

and representation (8) follows from thisFinally let us note that the uniqueness up to a unitary

transformation is obvious from the square root relation(12)

4 International Journal of Stochastic Analysis

Remark 13 The Fredholm representation (8) holds also forinfinite intervals that is 119879 = infin if the trace condition (9)holds Unfortunately this is seldom the case

Remark 14 The above proof shows that the Fredholm repre-sentation (8) holds in law However one can also constructthe process119883 via (8) for a given Brownian motion119882 In thiscase representation (8) holds of course in 1198712 Finally notethat in general it is not possible to construct the Brownianmotion in representation (8) from the process 119883 Indeedthere might not be enough randomness in 119883 To construct119882 from119883 one needs that the indicators 1

119905 119905 isin [0 119879] belong

to the range of the operator119870119879

Remark 15 We remark that the separability of 119883 ensuresrepresentation of form (8) where the kernel 119870

119879only satisfies

a weaker condition 119870119879(119905 sdot) isin 119871

2([0 119879]) for all 119905 isin [0 119879]

which may happen if the trace condition (9) fails In thiscase however the associated operator119870

119879does not belong to

1198712([0 119879]) which may be undesirable

Example 16 Let us consider the following very degeneratecase suppose 119883

119905= 119891(119905)120585 where 119891 is deterministic and 120585 is a

standard normal random variable Suppose 119879 gt 1 Then

119883119905= int

119879

0

119891 (119905) 1[01)(119904) d119882s (15)

So 119870119879(119905 119904) = 119891(119905)1

[01)(119904) Now if 119891 isin 1198712([0 119879]) then

condition (9) is satisfied and 119870119879isin 119871

2([0 119879]

2) On the other

hand even if 119891 notin 1198712([0 119879]) we can still write 119883 in form(15) However in this case the kernel 119870

119879does not belong to

1198712([0 119879]

2)

Example 17 Consider a truncated series expansion

119883119905=

119899

sum

119896=1

119890119879

119896(119905) 120585

119896 (16)

where 120585119896are independent standard normal random variables

and

119890119879

119896(119905) = int

119905

0

119879

119896(119904) d119904 (17)

where 119879119896 119896 isin N is an orthonormal basis in 1198712([0 119879]) Now

it is straightforward to check that this process is not purelynondeterministic (see [15] for definition) and consequently119883 cannot have Volterra representation while it is clear that119883 admits a Fredholm representation On the other hand bychoosing the functions 119879

119896to be the trigonometric basis on

1198712([0 119879])119883 is a finite-rank approximation of the Karhunen-

Loeve representation of standard Brownianmotion on [0 119879]Hence by letting 119899 tend to infinity we obtain the standardBrownian motion and hence a Volterra process

Example 18 Let119882 be a standard Brownian motion on [0 119879]and consider the Brownian bridge Now there are two rep-resentations of the Brownian bridge (see [16] and references

therein on the representations of Gaussian bridges) Theorthogonal representation is

119861119905= 119882

119905minus119905

119879119882

119879 (18)

Consequently 119861 has a Fredholm representation with kernel119870119879(119905 119904) = 1

119905(119904) minus 119905119879 The canonical representation of the

Brownian bridge is

119861t = (119879 minus 119905) int119905

0

1

119879 minus 119904d119882

119904 (19)

Consequently the Brownian bridge has also a Volterra-typerepresentation with kernel 119870(119905 119904) = (119879 minus 119905)(119879 minus 119904)

4 Transfer Principle and Stochastic Analysis

41 Wiener Integrals Theorem 22 is the transfer principle inthe context of Wiener integrals The same principle extendsto multiple Wiener integrals and Malliavin calculus later inthe following subsections

Recall that for any kernel Γ isin 1198712([0 119879]2) its associatedoperator on 1198712([0 119879]) is

Γ119891 (119905) = int

119879

0

119891 (119904) Γ (119905 119904) d119904 (20)

Definition 19 (adjoint associated operator) The adjoint asso-ciated operator Γlowast of a kernel Γ isin 1198712([0 119879]2) is defined bylinearly extending the relation

Γlowast1

119905= Γ (119905 sdot) (21)

Remark 20 Thename andnotation of ldquoadjointrdquo for119870lowast

119879comes

from Alos et al [1] where they showed that in their Volterracontext119870lowast

119879admits a kernel and is an adjoint of119870

119879in the sense

that

int

119879

0

119870lowast

119879119891 (119905) 119892 (119905) d119905 = int

119879

0

119891 (119905)119870119879119892 (d119905) (22)

for step-functions 119891 and 119892 belonging to 1198712([0 119879]) It isstraightforward to check that this statement is valid also inour case

Example 21 Suppose the kernel Γ(sdot 119904) is of bounded variationfor all 119904 and that 119891 is nice enough Then

Γlowast119891 (119904) = int

119879

0

119891 (119905) Γ (d119905 119904) (23)

Theorem 22 (transfer principle for Wiener integrals) Let 119883be a separable centered Gaussian process with representation(8) and let 119891 isinH

119879 Then

int

119879

0

119891 (119905) d119883119905= int

119879

0

119870lowast

119879119891 (119905) d119882

119905 (24)

Proof Assume first that 119891 is an elementary function of form

119891 (119905) =

119899

sum

119896=1

1198861198961119860119896

(25)

International Journal of Stochastic Analysis 5

for some disjoint intervals 119860119896= (119905

119896minus1 119905119896] Then the claim

follows by the very definition of the operator119870lowast

119879andWiener

integral with respect to 119883 together with representation(8) Furthermore this shows that 119870lowast

119879provides an isometry

between H119879and 1198712([0 119879]) Hence H

119879can be viewed as a

closure of elementary functions with respect to 119891H119879 =119870

lowast

119879119891

1198712([0119879])

which proves the claim

42 Multiple Wiener Integrals The study of multiple Wienerintegrals goes back to Ito [17] who studied the case ofBrownian motion Later Huang and Cambanis [18] extendedthe notion to general Gaussian processes Dasgupta andKallianpur [19 20] and Perez-Abreu and Tudor [21] studiedmultiple Wiener integrals in the context of fractional Brown-ian motion In [19 20] a method that involved a prior controlmeasure was used and in [21] a transfer principle was usedOur approach here extends the transfer principle methodused in [21]

We begin by recalling multiple Wiener integrals withrespect to Brownian motion and then we apply transferprinciple to generalize the theory to arbitrary Gaussianprocess

Let 119891 be an elementary function on [0 119879]119901 that vanisheson the diagonals that is

119891 =

119899

sum

1198941 119894119901=1

1198861198941 sdotsdotsdot119894119901

1Δ 1198941

timessdotsdotsdottimesΔ 119894119901 (26)

whereΔ119896fl [119905

119896minus1 119905119896) and 119886

1198941 119894119901= 0whenever 119894

119896= 119894

ℓfor some

119896 = ℓ For such 119891 we define the multiple Wiener integral as

119868119882

119879119901(119891) fl int

119879

0

sdot sdot sdot int

119879

0

119891 (1199051 119905

119901) 120575119882

1199051sdot sdot sdot 120575119882

119905119901

fl119899

sum

1198941i119901=11198861198941 119894119901Δ119882

1199051sdot sdot sdot Δ119882

119905119901

(27)

where we have denoted Δ119882119905119896fl119882

119905119896minus119882

119905119896minus1 For 119901 = 0 we set

119868119882

0(119891) = 119891 Now it can be shown that elementary functions

that vanish on the diagonals are dense in 1198712([0 119879]119901)Thusone can extend the operator 119868119882

119879119901to the space 1198712([0 119879]119901) This

extension is called the multiple Wiener integral with respectto the Brownian motion

Remark 23 It is well-known that 119868119882119879119901(119891) can be under-

stood as multiple or iterated Ito integrals if and only if119891(119905

1 119905

119901) = 0 unless 119905

1le sdot sdot sdot le 119905

119901 In this case we have

119868119882

119879119901(119891)

= 119901 int

119879

0

int

119905119901

0

sdot sdot sdot int

1199052

0

119891 (1199051 119905

119901) d119882

1199051sdot sdot sdot d119882

119905119901

(28)

For the case of Gaussian processes that are not martingalesthis fact is totally useless

For a general Gaussian process119883 recall first the Hermitepolynomials

119867119901(119909) fl

(minus1)119901

119901119890(12)119909

2 d119901

d119909119901(119890

minus(12)1199092

) (29)

For any 119901 ge 1 let the 119901th Wiener chaos of 119883 be theclosed linear subspace of 1198712(Ω) generated by the randomvariables 119867

119901(119883(120593)) 120593 isin H 120593H = 1 where 119867119901

is the119901th Hermite polynomial It is well-known that the mapping119868119883

119901(120593

otimes119901) = 119901119867

119901(119883(120593)) provides a linear isometry between

the symmetric tensor product H⊙119901 and the 119901th Wienerchaos The random variables 119868119883

119901(120593

otimes119901) are called multiple

Wiener integrals of order 119901 with respect to the Gaussianprocess119883

Let us now consider the multipleWiener integrals 119868119879119901

fora general Gaussian process119883 We define the multiple integral119868119879119901

by using the transfer principle in Definition 25 and laterargue that this is the ldquocorrectrdquo way of defining them So let119883 be a centered Gaussian process on [0 119879]with covariance 119877and representation (8) with kernel 119870

119879

Definition 24 (119901-fold adjoint associated operator) Let119870119879be

the kernel in (8) and let119870lowast

119879be its adjoint associated operator

Define

119870lowast

119879119901fl (119870lowast

119879)otimes119901

(30)

In the same way define

H119879119901

fl Hotimes119901

119879

H0

119879119901fl (Hotimes119901

119879)0

(31)

Here the tensor products are understood in the sense ofHilbert spaces that is they are closed under the inner productcorresponding to the 119901-fold product of the underlying innerproduct

Definition 25 Let 119883 be a centered Gaussian process withrepresentation (8) and let 119891 isinH

119879119901 Then

119868119879119901(119891) fl 119868119882

119879119901(119870

lowast

119879119901119891) (32)

The following example should convince the reader thatthis is indeed the correct definition

Example 26 Let 119901 = 2 and let ℎ = ℎ1otimesℎ

2 where both ℎ

1and

ℎ2are step-functions Then

(119870lowast

1198792ℎ) (119909 119910) = (119870

lowast

119879ℎ1) (119909) (119870

lowast

119879ℎ2) (119910)

119868119882

1198792(119870

lowast

1198792119891)

= int

119879

0

119870lowast

119879ℎ1(V) d119882V

sdot int

119879

0

119870lowast

119879ℎ2(119906) d119882

119906minus ⟨119870

lowast

119879ℎ1 119870

lowast

119879ℎ2⟩1198712([0119879])

= 119883 (ℎ1)119883 (ℎ

2) minus ⟨ℎ

1 ℎ

2⟩H119879

(33)

6 International Journal of Stochastic Analysis

The following proposition shows that our approach todefine multiple Wiener integrals is consistent with the tra-ditional approach where multiple Wiener integrals for moregeneral Gaussian process 119883 are defined as the closed linearspace generated by Hermite polynomials

Proposition 27 Let 119867119901be the 119901th Hermite polynomial and

let ℎ isinH119879 Then

119868119879119901(ℎ

otimes119901) = 119901 ℎ

119901

H119879119867119901(119883 (ℎ)

ℎH119879

) (34)

Proof First note that without loss of generalitywe can assumeℎH119879

= 1Now by the definition of multipleWiener integralwith respect to119883 we have

119868119879119901(ℎ

otimes119901) = 119868

119882

119879119901(119870

lowast

119879119901ℎotimes119901) (35)

where

119870lowast

119879119901ℎotimes119901= (119870

lowast

119879ℎ)

otimes119901

(36)

Consequently by [22 Proposition 114] we obtain

119868119879119901(ℎ

otimes119901) = 119901119867

119901(119882 (119870

lowast

119879ℎ)) (37)

which implies the result together withTheorem 22

Proposition 27 extends to the following product formulawhich is also well-known in the Gaussianmartingale case butapparently new for general Gaussian processes Again theproof is straightforward application of transfer principle

Proposition 28 Let 119891 isinH119879119901

and 119892 isinH119879119902Then

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimes

119870119879119903119892)

(38)

where

119891otimes119870119879119903119892 = (119870

lowast

119879119901+119902minus2119903)minus1

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (39)

and (119870lowast

119879119901+119902minus2119903)minus1 denotes the preimage of 119870lowast

119879119901+119902minus2119903

Proof The proof follows directly from the definition of119868119879119901(119891) and [22 Proposition 113]

Example 29 Let 119891 isin H119879119901

and 119892 isin H119879119902

be of forms119891(119909

1 119909

119901) = prod

119901

119896=1119891119896(119909

119896) and119892(119910

1 119910

119901) = prod

119902

119896=1119892119896(119910

119896)

Then

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (119909

1 119909

119901minus119903 119910

1 119910

119902minus119903)

= int[0119879]119903

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119903

prod

119895=1

119870lowast

119879119891119901minus119903+119895

(119904119895)

sdot

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot

119903

prod

119895=1

119870lowast

119879119892119902minus119903+119895(119904

119895) d119904

1sdot sdot sdot d119904

119903

=

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot ⟨

119903

prod

119895=1

119891119901minus119903+119895

119903

prod

119895=1

119892119902minus119903+119895⟩

H119879119903

(40)

Hence119891otimes

119870119879119903119892

=

119901minus119903

prod

119896=1

119891119896(119909

119896)

119902minus119903

prod

119896=1

119892119896(119910

119896)⟨

119903

prod

119895=1

119891119895

119903

prod

119895=1

119892119895⟩

H119879119903

(41)

Remark 30 In the literature multiple Wiener integrals areusually defined as the closed linear space spanned byHermitepolynomials In such a case Proposition 27 is clearly true bythe very definition Furthermore one has a multiplicationformula (see eg [23])

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimesH119879119903

119892)

(42)

where 119891otimesH119879119903119892 denotes symmetrization of tensor product

119891otimesH119879119903119892 =

infin

sum

1198941 119894119903=1

⟨119891 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

otimes ⟨119892 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

(43)

and 119890119896 119896 = 1 is a complete orthonormal basis of the

Hilbert space H119879 Clearly by Proposition 27 both formulas

coincide This also shows that (39) is well-defined

43 Malliavin Calculus and Skorohod Integrals We begin byrecalling some basic facts on Malliavin calculus

Definition 31 Denote by S the space of all smooth randomvariables of the form

119865 = 119891 (119883 (1205931) 119883 (120593

119899)) 120593

1 120593

119899isinH

119879 (44)

International Journal of Stochastic Analysis 7

where 119891 isin 119862infin119887(R119899) that is 119891 and all its derivatives are

bounded The Malliavin derivative 119863119879119865 = 119863

119883

119879119865 of 119865 is an

element of 1198712(ΩH119879) defined by

119863119879119865 =

119899

sum

119894=1

120597119894119891 (119883 (120593

1) 119883 (120593

119899)) 120593

119894 (45)

In particular119863119879119883119905= 1

119905

Definition 32 Let D12= D12

119883be the Hilbert space of all

square integrable Malliavin differentiable random variablesdefined as the closure of S with respect to norm

1198652

12= E [|119865|

2] + E [

10038171003817100381710038171198631198791198651003817100381710038171003817

2

H119879] (46)

The divergence operator 120575119879is defined as the adjoint

operator of the Malliavin derivative119863119879

Definition 33 The domain Dom 120575119879of the operator 120575

119879is the

set of random variables 119906 isin 1198712(ΩH119879) satisfying

10038161003816100381610038161003816E ⟨119863

119879119865 119906⟩

H119879

10038161003816100381610038161003816le 119888

119906119865

1198712 (47)

for any 119865 isin D12 and some constant 119888119906depending only on 119906

For 119906 isin Dom 120575119879the divergence operator 120575

119879(119906) is a square

integrable random variable defined by the duality relation

E [119865120575119879(119906)] = E ⟨119863

119879119865 119906⟩

H119879(48)

for all 119865 isin D12

Remark 34 It is well-known that D12sub Dom 120575

119879

We use the notation

120575119879(119906) = int

119879

0

119906119904120575119883

119904 (49)

Theorem 35 (transfer principle for Malliavin calculus) Let119883 be a separable centered Gaussian process with Fredholmrepresentation (8) Let 119863

119879and 120575

119879be the Malliavin derivative

and the Skorohod integral with respect to119883 on [0 119879] Similarlylet 119863119882

119879and 120575119882

119879be the Malliavin derivative and the Skorohod

integral with respect to the Brownianmotion119882 of (8) restrictedon [0 119879] Then

120575119879= 120575

119882

119879119870lowast

119879

119870lowast

119879119863119879= 119863

119882

119879

(50)

Proof The proof follows directly from transfer principle andthe isometry provided by 119870lowast

119879with the same arguments as in

[1] Indeed by isometry we have

H119879= (119870

lowast

119879)minus1

(1198712([0 119879])) (51)

where (119870lowast

119879)minus1 denotes the preimage which implies that

D12(H

119879) = (119870

lowast

119879)minus1

(D12

119882(119871

2([0 119879]))) (52)

which justifies 119870lowast

119879119863119879= 119863

119882

119879 Furthermore we have relation

E ⟨119906119863119879119865⟩

H119879= E ⟨119870

lowast

119879119906119863

119882

119879119865⟩

1198712([0119879])

(53)

for any smooth random variable 119865 and 119906 isin 1198712(ΩH119879)

Hence by the very definition ofDom 120575 and transfer principlewe obtain

Dom 120575119879= (119870

lowast

119879)minus1

(Dom 120575119882119879) (54)

and 120575119879(119906) = 120575

119882

119879(119870

lowast

119879119906) proving the claim

Now we are ready to show that the definition of themultiple Wiener integral 119868

119879119901in Section 42 is correct in the

sense that it agrees with the iterated Skorohod integral

Proposition 36 Let ℎ isin H119879119901

be of form ℎ(1199091 119909

119901) =

prod119901

119896=1ℎ119896(119909

119896) Then ℎ is iteratively 119901 times Skorohod integrable

and

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901) 120575119883

1199051sdot sdot sdot 120575119883

119905119901= 119868

119879119901(ℎ) (55)

Moreover if ℎ isin H0

119879119901is such that it is 119901 times iteratively

Skorohod integrable then (55) still holds

Proof Again the idea is to use the transfer principle togetherwith induction Note first that the statement is true for 119901 = 1by definition and assume next that the statement is validfor 119896 = 1 119901 We denote 119891

119895= prod

119895

119896=1ℎ119896(119909

119896) Hence by

induction assumption we have

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901 119905V) 1205751198831199051

sdot sdot sdot 120575119883119905119901120575119883V

= int

119879

0

119868119879119901(119891

119901) ℎ

119901+1(V) 120575119883V

(56)

Put now 119865 = 119868119879119901(119891

119901) and 119906(119905) = ℎ

119901+1(119905) Hence by [22

Proposition 133] and by applying the transfer principle weobtain that 119865119906 belongs to Dom 120575

119879and

120575119879(119865119906) = 120575

119879(119906) 119865 minus ⟨119863

119905119865 119906 (119905)⟩

H119879

= 119868119882

119879(119870

lowast

119879ℎ119901+1) 119868

119882

119879119901(119870

lowast

119879119901119891119901)

minus 119901 ⟨119868119879119901minus1(119891

119901(sdot 119905)) ℎ

119901+1(119905)⟩

H119879

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119882

119879119901minus1(119870

lowast

119879119901minus1119891119901otimes1119870lowast

119879ℎ119901+1)

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119879119901minus1(119891

119901otimes1198701198791ℎ119901+1)

(57)

Hence the result is valid also for 119901+ 1 by Proposition 28 with119902 = 1

The claim for general ℎ isinH0

119879119901follows by approximating

with a product of simple function

8 International Journal of Stochastic Analysis

Remark 37 Note that (55) does not hold for arbitrary ℎ isinH0

119879119901in general without the a priori assumption of 119901 times

iterative Skorohod integrability For example let 119901 = 2 and119883 = 119861

119867 be a fractional Brownian motion with119867 le 14 anddefine ℎ

119905(119904 V) = 1

119905(119904)1

119904(V) for some fixed 119905 isin [0 119879] Then

int

119879

0

ℎ119905(119904 V) 120575119883V = 119883119904

1119905(119904) (58)

But119883sdot1119905does not belong to Dom 120575

119879(see [7])

We end this section by providing an extension of Ito for-mulas provided by Alos et al [1] They considered GaussianVolterra processes that is they assumed the representation

119883119905= int

119905

0

119870 (119905 119904) d119882119904 (59)

where the kernel119870 satisfied certain technical assumptions In[1] it was proved that in the case of Volterra processes one has

119891 (119883119905) = 119891 (0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(60)

if 119891 satisfies the growth condition

max [1003816100381610038161003816119891 (119909)1003816100381610038161003816 100381610038161003816100381610038161198911015840(119909)100381610038161003816100381610038161003816100381610038161003816100381611989110158401015840(119909)10038161003816100381610038161003816] le 119888119890

120582|119909|2

(61)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1 In the

following we will consider different approach which enablesus to

(i) prove that such formula holds with minimal require-ments

(ii) give more instructive proof of such result(iii) extend the result from Volterra context to more

general Gaussian processes(iv) drop some technical assumptions posed in [1]

For simplicity we assume that the variance of 119883 is ofbounded variation to guarantee the existence of the integral

int

119879

0

11989110158401015840(119883

119905) d119877 (119905 119905) (62)

If the variance is not of bounded variation then integral (62)may be understood by integration by parts if 11989110158401015840 is smoothenough or in the general case via the inner product ⟨sdot sdot⟩H119879 InTheorem 40 we also have to assume that the variance of119883is bounded

The result for polynomials is straightforward once weassume that the paths of polynomials of 119883 belong to1198712(ΩH

119879)

Proposition 38 (Ito formula for polynomials) Let 119883 be aseparable centered Gaussian process with covariance 119877 andassume that 119901 is a polynomial Furthermore assume that for

each polynomial 119901 one has 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) Then for

each 119905 isin [0 119879] one has

119901 (119883119905) = 119901 (119883

0) + int

119905

0

1199011015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11990110158401015840(119883

119904) d119877 (119904 119904)

(63)

if and only if119883sdot1119905belongs to Dom 120575

119879

Remark 39 The message of the above result is that once theprocesses 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) then they automatically

belong to the domain of 120575119879which is a subspace of 1198712(ΩH

119879)

However in order to check 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) one needs

more information on the kernel 119870119879 A sufficient condition is

provided in Corollary 43 which covers many cases of interest

Proof By definition and applying transfer principle we haveto prove that 1199011015840(119883

sdot)1

119905belongs to domain of 120575

119879and that

Eint119905

0

119863119904119866119870

lowast

119879[119901

1015840(119883

sdot) 1

119905] d119904

= E [119866119901 (119883119905)] minus E [119866119901 (119883

0)]

minus1

2int

119905

0

E [11986611990110158401015840(119883

119905)] d119877 (119905 119905)

(64)

for every random variable 119866 from a total subset of 1198712(Ω)In other words it is sufficient to show that (69) is valid forrandom variables of form 119866 = 119868119882

119899(ℎ

otimes119899) where ℎ is a step-

functionNote first that it is sufficient to prove the claim only for

Hermite polynomials119867119896 119896 = 1 Indeed it is well-known

that any polynomial can be expressed as a linear combinationof Hermite polynomials and consequently the result forarbitrary polynomial 119901 follows by linearity

We proceed by induction First it is clear that firsttwo polynomials 119867

0and 119867

1satisfy (64) Furthermore by

assumption 1198671015840

2(119883

sdot)1

119905belongs to Dom 120575

119879from which (64)

is easily deduced by [22 Proposition 133] Assume nextthat the result is valid for Hermite polynomials 119867

119896 119896 =

0 1 119899 Then recall well-known recursion formulas

119867119899+1(119909) = 119909119867

119899(119909) minus 119899119867

119899minus1(119909)

1198671015840

119899(119909) = 119899119867

119899minus1(119909)

(65)

The induction step follows with straightforward calcula-tions by using the recursion formulas above and [22Proposition 133] We leave the details to the reader

We will now illustrate how the result can be generalizedfor functions satisfying the growth condition (61) by usingProposition 38 First note that the growth condition (61) isindeed natural since it guarantees that the left side of (60)is square integrable Consequently since operator 120575

119879is a

mapping from 1198712(ΩH119879) into 1198712(Ω) functions satisfying

(61) are the largest class of functions for which (60) can holdHowever it is not clear in general whether 1198911015840(119883

sdot)1

119905belongs

International Journal of Stochastic Analysis 9

to Dom 120575119879 Indeed for example in [1] the authors posed

additional conditions on the Volterra kernel 119870 to guaranteethis As our main result we show that E1198911015840(119883

sdot)1

1199052

H119879lt infin

implies that (60) holds In other words not only is the Itoformula (60) natural but it is also the only possibility

Theorem 40 (Ito formula for Skorohod integrals) Let 119883 bea separable centered Gaussian process with covariance 119877 suchthat all the polynomials 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) Assume that

119891 isin 1198622 satisfies growth condition (61) and that the variance of

119883 is bounded and of bounded variation If

E100381710038171003817100381710038171198911015840(119883

sdot) 1

119905

10038171003817100381710038171003817

2

H119879lt infin (66)

for any 119905 isin [0 119879] then

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(67)

Proof In this proof we assume for notational simplicity andwith no loss of generality that sup

0le119904le119879119877(119904 119904) = 1

First it is clear that (66) implies that 1198911015840(119883sdot)1

119905belongs to

domain of 120575119879 Hence we only have to prove that

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E [119866119891 (119883119905)] minus E [119866119891 (119883

0)]

minus1

2int

119905

0

E [11986611989110158401015840(119883

119904)] d119877 (119904 119904)

(68)

for every random variable 119866 = 119868119882119899(ℎ

otimes119899)

Now it is well-known that Hermite polynomials whenproperly scaled form an orthogonal system in 1198712(R) whenequipped with the Gaussian measure Now each 119891 satisfyingthe growth condition (61) has a series representation

119891 (119909) =

infin

sum

119896=0

120572119896119867119896(119909) (69)

Indeed the growth condition (61) implies that

intR

100381610038161003816100381610038161198911015840(119909)10038161003816100381610038161003816

2

119890minus1199092(2sup

0le119904le119879119877(119904119904))d119909 lt infin (70)

Furthermore we have

119891 (119883119904) =

infin

sum

119896=0

120572119896119867119896(119883

119904) (71)

where the series converge almost surely and in 1198712(Ω) andsimilar conclusion is valid for derivatives1198911015840(119883

119904) and 11989110158401015840(119883

119904)

Then by applying (66) we obtain that for any 120598 gt 0 thereexists119873 = 119873

120598such that we have

E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598 119899 ge 119873 (72)

where

1198911015840

119899(119883

119904) =

infin

sum

119896=119899

120572119896119867

1015840

119896(119883

119904) (73)

Consequently for random variables of form 119866 = 119868119882119899(ℎ

otimes119899) we

obtain by choosing 119873 large enough and applying Proposi-tion 38 that

E [119866119891 (119883119905)] minus E [119866119891 (119883

0)] minus1

2

sdot int

119905

0

E (11986611989110158401015840(119883

119905)) d119877 (119905 119905)

minus E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598

(74)

Now the left side does not depend on 119899 which concludes theproof

Remark 41 Note that actually it is sufficient to have

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

=

infin

sum

119896=1

120572119896E ⟨119863

119879119866119867

1015840

119896(119883

sdot) 1

119905⟩H119879

(75)

from which the result follows by Proposition 38 Further-more taking account growth condition (61) this is actuallysufficient and necessary condition for formula (60) to holdConsequently our method can also be used to obtain Itoformulas by considering extended domain of 120575

119879(see [9] or

[11]) This is the topic of Section 44

Example 42 It is known that if 119883 = 119861119867 is a fractional Brow-nian motion with119867 gt 14 then 1198911015840(119883

sdot)1

119905satisfies condition

(66) while for 119867 le 14 it does not (see [22 Chapter 5])Consequently a simple application of Theorem 40 coversfractional Brownian motion with 119867 gt 14 For the case119867 le 14 one has to consider extended domain of 120575

119879which is

proved in [9] Consequently in this case we have (75) for any119865 isin S

We end this section by illustrating the power of ourmethod with the following simple corollary which is anextension of [1 Theorem 1]

Corollary 43 Let 119883 be a separable centered continuousGaussian process with covariance 119877 that is bounded such thatthe Fredholm kernel 119870

119879is of bounded variation and

int

119879

0

(int

119879

0

1003817100381710038171003817119883119905minus 119883

119904

10038171003817100381710038171198712(Ω)

10038161003816100381610038161198701198791003816100381610038161003816 (d119905 119904))

2

d119904 lt infin (76)

Then for any 119905 isin [0 119879] one has

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(77)

10 International Journal of Stochastic Analysis

Proof Note that assumption is a Fredholm version of con-dition (K2) in [1] which implies condition (66) Hence theresult follows byTheorem 40

44 Extended Divergence Operator As shown in Section 43the Ito formula (60) is the only possibility However theproblem is that the space 1198712(ΩH

119879) may be too small to

contain the elements 1198911015840(119883sdot)1

119905 In particular it may happen

that not even the process 119883 itself belongs to 1198712(ΩH119879) (see

eg [7] for the case of fractional Brownian motion with119867 le 14) This problem can be overcome by consideringan extended domain of 120575

119879 The idea of extended domain is

to extend the inner product ⟨119906 120593⟩H119879 for simple 120593 to moregeneral processes 119906 and then define extended domain by(47) with a restricted class of test variables 119865 This also givesanother intuitive reason why extended domain of 120575

119879can be

useful indeed here we have proved that Ito formula (60) isthe only possibility and what one essentially needs for suchresult is the following

(i) 119883sdot1119905belongs to Dom 120575

119879

(ii) Equation (75) is valid for functions satisfying (61)

Consequently one should look for extensions of operator 120575119879

such that these two things are satisfiedTo facilitate the extension of domain we make the

following relatively moderate assumption

(H) The function 119905 997891rarr 119877(119905 119904) is of bounded variation on[0 119879] and

sup119905isin[0119879]

int

119879

0

|119877| (d119904 119905) lt infin (78)

Remark 44 Note that we are making the assumption on thecovariance 119877 not the kernel 119870

119879 Hence our case is different

from that of [1] Also [11] assumed absolute continuity in 119877we are satisfied with bounded variation

We will follow the idea from Lei and Nualart [11] andextend the inner product ⟨sdot sdot⟩H119879 beyondH

119879

Consider a step-function 120593Then on the one hand by theisometry property we have

⟨120593 1119905⟩H119879= int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 (79)

where 119892119905(119904) = 119870(119905 119904) isin 119871

2([0 119879]) On the other hand by

using adjoint property (see Remark 20) we obtain

int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 = int

119879

0

120593 (119904) (119870119879119892119905) (d119904) (80)

where computing formally we have

(119870119879119892119905) (d119904) = int

119879

0

119892119905(119906)119870

119879(d119904 119906) d119906

= int

119879

0

119870119879(119905 119906)119870

119879(d119904 119906) d119906 = 119877 (119905 d119904)

(81)

Consequently

⟨120593 1119905⟩H119879= int

119879

0

120593 (119904) 119877 (119905 d119904) (82)

This gives motivation to the following definition similar tothat of [11 Definition 21]

Definition 45 Denote by T119879the space of measurable func-

tions 119892 satisfying

sup119905isin[0119879]

int

119879

0

1003816100381610038161003816119892 (119904)1003816100381610038161003816 |119877| (119905 d119904) lt infin (83)

and let 120593 be a step-function of form 120593 = sum119899119896=11198871198961119905119896 Then we

extend ⟨sdot sdot⟩H119879 toT119879by defining

⟨119892 120593⟩H119879=

119899

sum

119896=1

119887119896int

119879

0

119892 (119904) 119877 (119905119896 d119904) (84)

In particular this implies that for 119892 and 120593 as above we have

⟨1198921119905 120593⟩

H119879= int

119905

0

119892 (119904) d ⟨1119904 120593⟩

H119879 (85)

We define extended domain Dom119864120575119879similarly as in [11]

Definition 46 A process 119906 isin 1198711(ΩT119879) belongs to Dom119864

120575119879

if10038161003816100381610038161003816E ⟨119906119863119865⟩H119879

10038161003816100381610038161003816le 119888

119906119865

2(86)

for any smooth random variable 119865 isin S In this case 120575(119906) isin1198712(Ω) is defined by duality relationship

E [119865120575 (119906)] = E ⟨119906119863119865⟩H119879 (87)

Remark 47 Note that in general Dom 120575119879and Dom119864

120575119879are

not comparable See [11] for discussion

Note now that if a function 119891 satisfies the growthcondition (61) then 1198911015840(119883

sdot)1

119905isin 119871

1(ΩT

119879) since (61) implies

E sup119905isin[0119879]

100381610038161003816100381610038161198911015840(119883

119905)10038161003816100381610038161003816

119901

lt infin (88)

for any 119901 lt (12120582)(sup119905isin[0119879]

119877(119905 119905))minus1 Consequently with

this definition we are able to get rid of the problem thatprocesses might not belong to corresponding H

119879-spaces

Furthermore this implies that the series expansion (69)converges in the norm 1198711(ΩT

119879) defined by

Eint119879

0

|119906 (119904)| |119877| (119905 d119904) (89)

which in turn implies (75) Hence it is straightforward toobtain the following by first showing the result for polynomi-als and then by approximating in a similar manner as done inSection 43 but using the extended domain instead

International Journal of Stochastic Analysis 11

Theorem 48 (Ito formula for extended Skorohod integrals)Let119883 be a separable centered Gaussian process with covariance119877 and assume that 119891 isin 1198622 satisfies growth condition (61)Furthermore assume that (H) holds and that the variance of119883is bounded and is of bounded variationThen for any 119905 isin [0 119879]the process 1198911015840(119883

sdot)1

119905belongs to Dom119864

120575119879and

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(90)

Remark 49 As an application of Theorem 48 it is straight-forward to derive version of Ito-Tanaka formula underadditional conditions which guarantee that for a certainsequence of functions 119891

119899we have the convergence of term

(12) int119905

011989110158401015840

119899(119883

119904)d119877(119904 119904) to the local time For details we refer

to [11] where authors derived such formula under theirassumptions

Finally let us note that the extension to functions 119891(119905 119909)is straightforward where 119891 satisfies the following growthcondition

max [1003816100381610038161003816119891 (119905 119909)1003816100381610038161003816 1003816100381610038161003816120597119905119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119909119891 (119905 119909)

1003816100381610038161003816]

le 119888119890120582|119909|2

(91)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1

Theorem 50 (Ito formula for extended Skorohod integralsII) Let 119883 be a separable centered Gaussian process withcovariance 119877 and assume that 119891 isin 11986212 satisfies growthcondition (91) Furthermore assume that (H) holds and thatthe variance of119883 is bounded and is of bounded variationThenfor any 119905 isin [0 119879] the process 120597

119909119891(sdot 119883

sdot)1

119905belongs to Dom119864

120575119879

and

119891 (119905 119883119905) = 119891 (0 119883

0) + int

119905

0

120597119909119891 (119904 119883

119904) 120575119883

119904

+ int

119905

0

120597119905119891 (119904 119883

119904) d119904

+1

2int

119905

0

120597119909119909119891 (119904 119883

119904) d119877 (119904 119904)

(92)

Proof Taking into account that we have no problems con-cerning processes to belong to the required spaces theformula follows by approximating with polynomials of form119901(119909)119902(119905) and following the proof of Theorem 40

5 Applications

We illustrate how some results transfer easily from theBrownian case to the Gaussian Fredholm processes

51 Equivalence in Law The transfer principle has alreadybeen used in connection with the equivalence of law of

Gaussian processes in for example [24] in the context offractional Brownian motions and in [2] in the context ofGaussianVolterra processes satisfying certain nondegeneracyconditions The following proposition uses the Fredholmrepresentation (8) to give a sufficient condition for theequivalence of general Gaussian processes in terms of theirFredholm kernels

Proposition 51 Let 119883 and be two Gaussian processes withFredholm kernels 119870

119879and

119879 respectively If there exists a

Volterra kernel ℓ isin 1198712([0 119879]2) such that

119879(119905 119904) = 119870

119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 (93)

then119883 and are equivalent in law

Proof Recall that by the Hitsuda representation theorem [5Theorem 63] a centered Gaussian process is equivalent inlaw to a Brownian motion on [0 119879] if and only if there existsa kernel ℓ isin 1198712([0 119879]2) and a Brownian motion119882 such that admits the representation

119905= 119882

119905minus int

119905

0

int

119904

0

ℓ (119904 119906) d119882119906d119904 (94)

Let119883 have the Fredholm representations

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (95)

Then is equivalent to119883 if it admits in law the representa-tion

119905

119889

= int

119879

0

119870119879(119905 119904) d

119904 (96)

where is connected to119882 of (95) by (94)In order to show (96) let

119905= int

119879

0

119879(119905 119904) d1198821015840

119905(97)

be the Fredholm representation of Here 1198821015840 is someBrownian motion Then by using connection (93) and theFubini theorem we obtain

119905= int

119879

0

119879(119905 119904) d1198821015840

119904

119889

= int

119879

0

119879(119905 119904) d119882

119904

= int

119879

0

(119870119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906) d119882

119904

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

International Journal of Stochastic Analysis 3

Definition 3 (associated operator) For a kernel Γ isin

1198712([0 119879]

2) one associates an operator on 1198712([0 119879]) also

denoted by Γ as

Γ119891 (119905) = int

119879

0

119891 (119904) Γ (119905 119904) d119904 (3)

Definition 4 (isonormal process) The isonormal processassociated with 119883 also denoted by 119883 is the Gaussian family(119883(ℎ) ℎ isin H

119879) where the Hilbert space H

119879= H

119879(119877) is

generated by the covariance 119877 as follows

(i) Indicators 1119905fl 1

[0119905) 119905 le 119879 belong toH

119879

(ii) H119879is endowed with the inner product ⟨1

119905 1

119904⟩H119879

fl119877(119905 119904)

Definition 4 states that 119883(ℎ) is the image of ℎ isin H119879in

the isometry that extends the relation

119883(1119905) fl 119883

119905(4)

linearly Consequently we can have the following definition

Definition 5 (Wiener integral) 119883(ℎ) is theWiener integral ofthe element ℎ isinH

119879with respect to119883 One will also denote

int

119879

0

ℎ (119905) d119883119905fl 119883 (ℎ) (5)

Remark 6 Eventually all the following will mean the same

119883(ℎ) = int

119879

0

ℎ (119905) d119883119905= int

119879

0

ℎ (119905) 120575119883119905= 119868

1198791(ℎ)

= 119868119879(ℎ)

(6)

Remark 7 TheHilbert spaceH119879is separable if and only if119883

is separable

Remark 8 Due to the completion under the inner product⟨sdot sdot⟩H119879

it may happen that the space H119879is not a space of

functions but contains distributions compare [13] for the caseof fractional Brownian motions with Hurst index bigger thanhalf

Definition 9 The function space H0

119879sub H

119879is the space

of functions that can be approximated by step-functions on[0 119879] in the inner product ⟨sdot sdot⟩H119879

Example 10 If the covariance 119877 is of bounded variation thenH0

119879is the space of functions 119891 satisfying

∬

119879

0

1003816100381610038161003816119891 (119905) 119891 (119904)1003816100381610038161003816 |119877| (d119904 d119905) lt infin (7)

Remark 11 Note that itmay be that119891 isinH0

119879but for some1198791015840 lt

119879 we have 11989111198791015840 notin H0

1198791015840 compare [14] for an example with

fractional Brownian motion with Hurst index less than halfFor this reason we keep the notation H

119879instead of simply

writing H For the same reason we include the dependenceof 119879 whenever there is one

3 Fredholm Representation

Theorem 12 (Fredholm representation) Let 119883 = (119883119905)119905isin[0119879]

be a separable centered Gaussian process Then there exists akernel119870

119879isin 119871

2([0 119879]

2) and a Brownian motion119882 = (119882

119905)119905ge0

independent of 119879 such that

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904(8)

in law if and only if the covariance 119877 of 119883 satisfies the tracecondition

int

119879

0

119877 (119905 119905) d119905 lt infin (9)

Representation (8) is unique in the sense that any otherrepresentation with kernel

119879 say is connected to (8) by

a unitary operator 119880 on 1198712([0 119879]) such that 119879= 119880119870

119879

Moreover one may assume that 119870119879is symmetric

Proof Let us first remark that (9) is precisely what we need toinvokeMercerrsquos theorem and take square root in the resultingexpansion

Now by Mercerrsquos theorem we can expand the covariancefunction 119877 on [0 119879]2 as

119877 (119905 119904) =

infin

sum

119894=1

120582119879

119894119890119879

119894(119905) 119890

119879

119894(119904) (10)

where (120582119879119894)infin

119894=1and (119890119879

119894)infin

119894=1are the eigenvalues and the eigen-

functions of the covariance operator

119877119879119891 (119905) = int

119879

0

119891 (119904) 119877 (119905 119904) d119904 (11)

Moreover (119890119879119894)infin

119894=1is an orthonormal system on 1198712([0 119879])

Now 119877119879 being a covariance operator admits a square

root operator119870119879defined by the relation

int

119879

0

119890119879

119894(119904) 119877

119879119890119879

119895(119904) d119904 = int

119879

0

119870119879119890119879

119894(119904) 119870

119879119890119879

119895(119904) d119904 (12)

for all 119890119879119894and 119890119879

119895 Now condition (9) means that 119877

119879is trace

class and consequently 119870119879is Hilbert-Schmidt operator In

particular 119870119879is a compact operator Therefore it admits

a kernel Indeed a kernel 119870119879can be defined by using the

Mercer expansion (10) as

119870119879(119905 119904) =

infin

sum

119894=1

radic120582119879

119894119890119879

119894(119905) 119890

119879

119894(119904) (13)

This kernel is obviously symmetric Now it follows that

119877 (119905 119904) = int

119879

0

119870119879(119905 119906)119870

119879(119904 119906) d119906 (14)

and representation (8) follows from thisFinally let us note that the uniqueness up to a unitary

transformation is obvious from the square root relation(12)

4 International Journal of Stochastic Analysis

Remark 13 The Fredholm representation (8) holds also forinfinite intervals that is 119879 = infin if the trace condition (9)holds Unfortunately this is seldom the case

Remark 14 The above proof shows that the Fredholm repre-sentation (8) holds in law However one can also constructthe process119883 via (8) for a given Brownian motion119882 In thiscase representation (8) holds of course in 1198712 Finally notethat in general it is not possible to construct the Brownianmotion in representation (8) from the process 119883 Indeedthere might not be enough randomness in 119883 To construct119882 from119883 one needs that the indicators 1

119905 119905 isin [0 119879] belong

to the range of the operator119870119879

Remark 15 We remark that the separability of 119883 ensuresrepresentation of form (8) where the kernel 119870

119879only satisfies

a weaker condition 119870119879(119905 sdot) isin 119871

2([0 119879]) for all 119905 isin [0 119879]

which may happen if the trace condition (9) fails In thiscase however the associated operator119870

119879does not belong to

1198712([0 119879]) which may be undesirable

Example 16 Let us consider the following very degeneratecase suppose 119883

119905= 119891(119905)120585 where 119891 is deterministic and 120585 is a

standard normal random variable Suppose 119879 gt 1 Then

119883119905= int

119879

0

119891 (119905) 1[01)(119904) d119882s (15)

So 119870119879(119905 119904) = 119891(119905)1

[01)(119904) Now if 119891 isin 1198712([0 119879]) then

condition (9) is satisfied and 119870119879isin 119871

2([0 119879]

2) On the other

hand even if 119891 notin 1198712([0 119879]) we can still write 119883 in form(15) However in this case the kernel 119870

119879does not belong to

1198712([0 119879]

2)

Example 17 Consider a truncated series expansion

119883119905=

119899

sum

119896=1

119890119879

119896(119905) 120585

119896 (16)

where 120585119896are independent standard normal random variables

and

119890119879

119896(119905) = int

119905

0

119879

119896(119904) d119904 (17)

where 119879119896 119896 isin N is an orthonormal basis in 1198712([0 119879]) Now

it is straightforward to check that this process is not purelynondeterministic (see [15] for definition) and consequently119883 cannot have Volterra representation while it is clear that119883 admits a Fredholm representation On the other hand bychoosing the functions 119879

119896to be the trigonometric basis on

1198712([0 119879])119883 is a finite-rank approximation of the Karhunen-

Loeve representation of standard Brownianmotion on [0 119879]Hence by letting 119899 tend to infinity we obtain the standardBrownian motion and hence a Volterra process

Example 18 Let119882 be a standard Brownian motion on [0 119879]and consider the Brownian bridge Now there are two rep-resentations of the Brownian bridge (see [16] and references

therein on the representations of Gaussian bridges) Theorthogonal representation is

119861119905= 119882

119905minus119905

119879119882

119879 (18)

Consequently 119861 has a Fredholm representation with kernel119870119879(119905 119904) = 1

119905(119904) minus 119905119879 The canonical representation of the

Brownian bridge is

119861t = (119879 minus 119905) int119905

0

1

119879 minus 119904d119882

119904 (19)

Consequently the Brownian bridge has also a Volterra-typerepresentation with kernel 119870(119905 119904) = (119879 minus 119905)(119879 minus 119904)

4 Transfer Principle and Stochastic Analysis

41 Wiener Integrals Theorem 22 is the transfer principle inthe context of Wiener integrals The same principle extendsto multiple Wiener integrals and Malliavin calculus later inthe following subsections

Recall that for any kernel Γ isin 1198712([0 119879]2) its associatedoperator on 1198712([0 119879]) is

Γ119891 (119905) = int

119879

0

119891 (119904) Γ (119905 119904) d119904 (20)

Definition 19 (adjoint associated operator) The adjoint asso-ciated operator Γlowast of a kernel Γ isin 1198712([0 119879]2) is defined bylinearly extending the relation

Γlowast1

119905= Γ (119905 sdot) (21)

Remark 20 Thename andnotation of ldquoadjointrdquo for119870lowast

119879comes

from Alos et al [1] where they showed that in their Volterracontext119870lowast

119879admits a kernel and is an adjoint of119870

119879in the sense

that

int

119879

0

119870lowast

119879119891 (119905) 119892 (119905) d119905 = int

119879

0

119891 (119905)119870119879119892 (d119905) (22)

for step-functions 119891 and 119892 belonging to 1198712([0 119879]) It isstraightforward to check that this statement is valid also inour case

Example 21 Suppose the kernel Γ(sdot 119904) is of bounded variationfor all 119904 and that 119891 is nice enough Then

Γlowast119891 (119904) = int

119879

0

119891 (119905) Γ (d119905 119904) (23)

Theorem 22 (transfer principle for Wiener integrals) Let 119883be a separable centered Gaussian process with representation(8) and let 119891 isinH

119879 Then

int

119879

0

119891 (119905) d119883119905= int

119879

0

119870lowast

119879119891 (119905) d119882

119905 (24)

Proof Assume first that 119891 is an elementary function of form

119891 (119905) =

119899

sum

119896=1

1198861198961119860119896

(25)

International Journal of Stochastic Analysis 5

for some disjoint intervals 119860119896= (119905

119896minus1 119905119896] Then the claim

follows by the very definition of the operator119870lowast

119879andWiener

integral with respect to 119883 together with representation(8) Furthermore this shows that 119870lowast

119879provides an isometry

between H119879and 1198712([0 119879]) Hence H

119879can be viewed as a

closure of elementary functions with respect to 119891H119879 =119870

lowast

119879119891

1198712([0119879])

which proves the claim

42 Multiple Wiener Integrals The study of multiple Wienerintegrals goes back to Ito [17] who studied the case ofBrownian motion Later Huang and Cambanis [18] extendedthe notion to general Gaussian processes Dasgupta andKallianpur [19 20] and Perez-Abreu and Tudor [21] studiedmultiple Wiener integrals in the context of fractional Brown-ian motion In [19 20] a method that involved a prior controlmeasure was used and in [21] a transfer principle was usedOur approach here extends the transfer principle methodused in [21]

We begin by recalling multiple Wiener integrals withrespect to Brownian motion and then we apply transferprinciple to generalize the theory to arbitrary Gaussianprocess

Let 119891 be an elementary function on [0 119879]119901 that vanisheson the diagonals that is

119891 =

119899

sum

1198941 119894119901=1

1198861198941 sdotsdotsdot119894119901

1Δ 1198941

timessdotsdotsdottimesΔ 119894119901 (26)

whereΔ119896fl [119905

119896minus1 119905119896) and 119886

1198941 119894119901= 0whenever 119894

119896= 119894

ℓfor some

119896 = ℓ For such 119891 we define the multiple Wiener integral as

119868119882

119879119901(119891) fl int

119879

0

sdot sdot sdot int

119879

0

119891 (1199051 119905

119901) 120575119882

1199051sdot sdot sdot 120575119882

119905119901

fl119899

sum

1198941i119901=11198861198941 119894119901Δ119882

1199051sdot sdot sdot Δ119882

119905119901

(27)

where we have denoted Δ119882119905119896fl119882

119905119896minus119882

119905119896minus1 For 119901 = 0 we set

119868119882

0(119891) = 119891 Now it can be shown that elementary functions

that vanish on the diagonals are dense in 1198712([0 119879]119901)Thusone can extend the operator 119868119882

119879119901to the space 1198712([0 119879]119901) This

extension is called the multiple Wiener integral with respectto the Brownian motion

Remark 23 It is well-known that 119868119882119879119901(119891) can be under-

stood as multiple or iterated Ito integrals if and only if119891(119905

1 119905

119901) = 0 unless 119905

1le sdot sdot sdot le 119905

119901 In this case we have

119868119882

119879119901(119891)

= 119901 int

119879

0

int

119905119901

0

sdot sdot sdot int

1199052

0

119891 (1199051 119905

119901) d119882

1199051sdot sdot sdot d119882

119905119901

(28)

For the case of Gaussian processes that are not martingalesthis fact is totally useless

For a general Gaussian process119883 recall first the Hermitepolynomials

119867119901(119909) fl

(minus1)119901

119901119890(12)119909

2 d119901

d119909119901(119890

minus(12)1199092

) (29)

For any 119901 ge 1 let the 119901th Wiener chaos of 119883 be theclosed linear subspace of 1198712(Ω) generated by the randomvariables 119867

119901(119883(120593)) 120593 isin H 120593H = 1 where 119867119901

is the119901th Hermite polynomial It is well-known that the mapping119868119883

119901(120593

otimes119901) = 119901119867

119901(119883(120593)) provides a linear isometry between

the symmetric tensor product H⊙119901 and the 119901th Wienerchaos The random variables 119868119883

119901(120593

otimes119901) are called multiple

Wiener integrals of order 119901 with respect to the Gaussianprocess119883

Let us now consider the multipleWiener integrals 119868119879119901

fora general Gaussian process119883 We define the multiple integral119868119879119901

by using the transfer principle in Definition 25 and laterargue that this is the ldquocorrectrdquo way of defining them So let119883 be a centered Gaussian process on [0 119879]with covariance 119877and representation (8) with kernel 119870

119879

Definition 24 (119901-fold adjoint associated operator) Let119870119879be

the kernel in (8) and let119870lowast

119879be its adjoint associated operator

Define

119870lowast

119879119901fl (119870lowast

119879)otimes119901

(30)

In the same way define

H119879119901

fl Hotimes119901

119879

H0

119879119901fl (Hotimes119901

119879)0

(31)

Here the tensor products are understood in the sense ofHilbert spaces that is they are closed under the inner productcorresponding to the 119901-fold product of the underlying innerproduct

Definition 25 Let 119883 be a centered Gaussian process withrepresentation (8) and let 119891 isinH

119879119901 Then

119868119879119901(119891) fl 119868119882

119879119901(119870

lowast

119879119901119891) (32)

The following example should convince the reader thatthis is indeed the correct definition

Example 26 Let 119901 = 2 and let ℎ = ℎ1otimesℎ

2 where both ℎ

1and

ℎ2are step-functions Then

(119870lowast

1198792ℎ) (119909 119910) = (119870

lowast

119879ℎ1) (119909) (119870

lowast

119879ℎ2) (119910)

119868119882

1198792(119870

lowast

1198792119891)

= int

119879

0

119870lowast

119879ℎ1(V) d119882V

sdot int

119879

0

119870lowast

119879ℎ2(119906) d119882

119906minus ⟨119870

lowast

119879ℎ1 119870

lowast

119879ℎ2⟩1198712([0119879])

= 119883 (ℎ1)119883 (ℎ

2) minus ⟨ℎ

1 ℎ

2⟩H119879

(33)

6 International Journal of Stochastic Analysis

The following proposition shows that our approach todefine multiple Wiener integrals is consistent with the tra-ditional approach where multiple Wiener integrals for moregeneral Gaussian process 119883 are defined as the closed linearspace generated by Hermite polynomials

Proposition 27 Let 119867119901be the 119901th Hermite polynomial and

let ℎ isinH119879 Then

119868119879119901(ℎ

otimes119901) = 119901 ℎ

119901

H119879119867119901(119883 (ℎ)

ℎH119879

) (34)

Proof First note that without loss of generalitywe can assumeℎH119879

= 1Now by the definition of multipleWiener integralwith respect to119883 we have

119868119879119901(ℎ

otimes119901) = 119868

119882

119879119901(119870

lowast

119879119901ℎotimes119901) (35)

where

119870lowast

119879119901ℎotimes119901= (119870

lowast

119879ℎ)

otimes119901

(36)

Consequently by [22 Proposition 114] we obtain

119868119879119901(ℎ

otimes119901) = 119901119867

119901(119882 (119870

lowast

119879ℎ)) (37)

which implies the result together withTheorem 22

Proposition 27 extends to the following product formulawhich is also well-known in the Gaussianmartingale case butapparently new for general Gaussian processes Again theproof is straightforward application of transfer principle

Proposition 28 Let 119891 isinH119879119901

and 119892 isinH119879119902Then

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimes

119870119879119903119892)

(38)

where

119891otimes119870119879119903119892 = (119870

lowast

119879119901+119902minus2119903)minus1

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (39)

and (119870lowast

119879119901+119902minus2119903)minus1 denotes the preimage of 119870lowast

119879119901+119902minus2119903

Proof The proof follows directly from the definition of119868119879119901(119891) and [22 Proposition 113]

Example 29 Let 119891 isin H119879119901

and 119892 isin H119879119902

be of forms119891(119909

1 119909

119901) = prod

119901

119896=1119891119896(119909

119896) and119892(119910

1 119910

119901) = prod

119902

119896=1119892119896(119910

119896)

Then

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (119909

1 119909

119901minus119903 119910

1 119910

119902minus119903)

= int[0119879]119903

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119903

prod

119895=1

119870lowast

119879119891119901minus119903+119895

(119904119895)

sdot

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot

119903

prod

119895=1

119870lowast

119879119892119902minus119903+119895(119904

119895) d119904

1sdot sdot sdot d119904

119903

=

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot ⟨

119903

prod

119895=1

119891119901minus119903+119895

119903

prod

119895=1

119892119902minus119903+119895⟩

H119879119903

(40)

Hence119891otimes

119870119879119903119892

=

119901minus119903

prod

119896=1

119891119896(119909

119896)

119902minus119903

prod

119896=1

119892119896(119910

119896)⟨

119903

prod

119895=1

119891119895

119903

prod

119895=1

119892119895⟩

H119879119903

(41)

Remark 30 In the literature multiple Wiener integrals areusually defined as the closed linear space spanned byHermitepolynomials In such a case Proposition 27 is clearly true bythe very definition Furthermore one has a multiplicationformula (see eg [23])

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimesH119879119903

119892)

(42)

where 119891otimesH119879119903119892 denotes symmetrization of tensor product

119891otimesH119879119903119892 =

infin

sum

1198941 119894119903=1

⟨119891 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

otimes ⟨119892 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

(43)

and 119890119896 119896 = 1 is a complete orthonormal basis of the

Hilbert space H119879 Clearly by Proposition 27 both formulas

coincide This also shows that (39) is well-defined

43 Malliavin Calculus and Skorohod Integrals We begin byrecalling some basic facts on Malliavin calculus

Definition 31 Denote by S the space of all smooth randomvariables of the form

119865 = 119891 (119883 (1205931) 119883 (120593

119899)) 120593

1 120593

119899isinH

119879 (44)

International Journal of Stochastic Analysis 7

where 119891 isin 119862infin119887(R119899) that is 119891 and all its derivatives are

bounded The Malliavin derivative 119863119879119865 = 119863

119883

119879119865 of 119865 is an

element of 1198712(ΩH119879) defined by

119863119879119865 =

119899

sum

119894=1

120597119894119891 (119883 (120593

1) 119883 (120593

119899)) 120593

119894 (45)

In particular119863119879119883119905= 1

119905

Definition 32 Let D12= D12

119883be the Hilbert space of all

square integrable Malliavin differentiable random variablesdefined as the closure of S with respect to norm

1198652

12= E [|119865|

2] + E [

10038171003817100381710038171198631198791198651003817100381710038171003817

2

H119879] (46)

The divergence operator 120575119879is defined as the adjoint

operator of the Malliavin derivative119863119879

Definition 33 The domain Dom 120575119879of the operator 120575

119879is the

set of random variables 119906 isin 1198712(ΩH119879) satisfying

10038161003816100381610038161003816E ⟨119863

119879119865 119906⟩

H119879

10038161003816100381610038161003816le 119888

119906119865

1198712 (47)

for any 119865 isin D12 and some constant 119888119906depending only on 119906

For 119906 isin Dom 120575119879the divergence operator 120575

119879(119906) is a square

integrable random variable defined by the duality relation

E [119865120575119879(119906)] = E ⟨119863

119879119865 119906⟩

H119879(48)

for all 119865 isin D12

Remark 34 It is well-known that D12sub Dom 120575

119879

We use the notation

120575119879(119906) = int

119879

0

119906119904120575119883

119904 (49)

Theorem 35 (transfer principle for Malliavin calculus) Let119883 be a separable centered Gaussian process with Fredholmrepresentation (8) Let 119863

119879and 120575

119879be the Malliavin derivative

and the Skorohod integral with respect to119883 on [0 119879] Similarlylet 119863119882

119879and 120575119882

119879be the Malliavin derivative and the Skorohod

integral with respect to the Brownianmotion119882 of (8) restrictedon [0 119879] Then

120575119879= 120575

119882

119879119870lowast

119879

119870lowast

119879119863119879= 119863

119882

119879

(50)

Proof The proof follows directly from transfer principle andthe isometry provided by 119870lowast

119879with the same arguments as in

[1] Indeed by isometry we have

H119879= (119870

lowast

119879)minus1

(1198712([0 119879])) (51)

where (119870lowast

119879)minus1 denotes the preimage which implies that

D12(H

119879) = (119870

lowast

119879)minus1

(D12

119882(119871

2([0 119879]))) (52)

which justifies 119870lowast

119879119863119879= 119863

119882

119879 Furthermore we have relation

E ⟨119906119863119879119865⟩

H119879= E ⟨119870

lowast

119879119906119863

119882

119879119865⟩

1198712([0119879])

(53)

for any smooth random variable 119865 and 119906 isin 1198712(ΩH119879)

Hence by the very definition ofDom 120575 and transfer principlewe obtain

Dom 120575119879= (119870

lowast

119879)minus1

(Dom 120575119882119879) (54)

and 120575119879(119906) = 120575

119882

119879(119870

lowast

119879119906) proving the claim

Now we are ready to show that the definition of themultiple Wiener integral 119868

119879119901in Section 42 is correct in the

sense that it agrees with the iterated Skorohod integral

Proposition 36 Let ℎ isin H119879119901

be of form ℎ(1199091 119909

119901) =

prod119901

119896=1ℎ119896(119909

119896) Then ℎ is iteratively 119901 times Skorohod integrable

and

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901) 120575119883

1199051sdot sdot sdot 120575119883

119905119901= 119868

119879119901(ℎ) (55)

Moreover if ℎ isin H0

119879119901is such that it is 119901 times iteratively

Skorohod integrable then (55) still holds

Proof Again the idea is to use the transfer principle togetherwith induction Note first that the statement is true for 119901 = 1by definition and assume next that the statement is validfor 119896 = 1 119901 We denote 119891

119895= prod

119895

119896=1ℎ119896(119909

119896) Hence by

induction assumption we have

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901 119905V) 1205751198831199051

sdot sdot sdot 120575119883119905119901120575119883V

= int

119879

0

119868119879119901(119891

119901) ℎ

119901+1(V) 120575119883V

(56)

Put now 119865 = 119868119879119901(119891

119901) and 119906(119905) = ℎ

119901+1(119905) Hence by [22

Proposition 133] and by applying the transfer principle weobtain that 119865119906 belongs to Dom 120575

119879and

120575119879(119865119906) = 120575

119879(119906) 119865 minus ⟨119863

119905119865 119906 (119905)⟩

H119879

= 119868119882

119879(119870

lowast

119879ℎ119901+1) 119868

119882

119879119901(119870

lowast

119879119901119891119901)

minus 119901 ⟨119868119879119901minus1(119891

119901(sdot 119905)) ℎ

119901+1(119905)⟩

H119879

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119882

119879119901minus1(119870

lowast

119879119901minus1119891119901otimes1119870lowast

119879ℎ119901+1)

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119879119901minus1(119891

119901otimes1198701198791ℎ119901+1)

(57)

Hence the result is valid also for 119901+ 1 by Proposition 28 with119902 = 1

The claim for general ℎ isinH0

119879119901follows by approximating

with a product of simple function

8 International Journal of Stochastic Analysis

Remark 37 Note that (55) does not hold for arbitrary ℎ isinH0

119879119901in general without the a priori assumption of 119901 times

iterative Skorohod integrability For example let 119901 = 2 and119883 = 119861

119867 be a fractional Brownian motion with119867 le 14 anddefine ℎ

119905(119904 V) = 1

119905(119904)1

119904(V) for some fixed 119905 isin [0 119879] Then

int

119879

0

ℎ119905(119904 V) 120575119883V = 119883119904

1119905(119904) (58)

But119883sdot1119905does not belong to Dom 120575

119879(see [7])

We end this section by providing an extension of Ito for-mulas provided by Alos et al [1] They considered GaussianVolterra processes that is they assumed the representation

119883119905= int

119905

0

119870 (119905 119904) d119882119904 (59)

where the kernel119870 satisfied certain technical assumptions In[1] it was proved that in the case of Volterra processes one has

119891 (119883119905) = 119891 (0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(60)

if 119891 satisfies the growth condition

max [1003816100381610038161003816119891 (119909)1003816100381610038161003816 100381610038161003816100381610038161198911015840(119909)100381610038161003816100381610038161003816100381610038161003816100381611989110158401015840(119909)10038161003816100381610038161003816] le 119888119890

120582|119909|2

(61)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1 In the

following we will consider different approach which enablesus to

(i) prove that such formula holds with minimal require-ments

(ii) give more instructive proof of such result(iii) extend the result from Volterra context to more

general Gaussian processes(iv) drop some technical assumptions posed in [1]

For simplicity we assume that the variance of 119883 is ofbounded variation to guarantee the existence of the integral

int

119879

0

11989110158401015840(119883

119905) d119877 (119905 119905) (62)

If the variance is not of bounded variation then integral (62)may be understood by integration by parts if 11989110158401015840 is smoothenough or in the general case via the inner product ⟨sdot sdot⟩H119879 InTheorem 40 we also have to assume that the variance of119883is bounded

The result for polynomials is straightforward once weassume that the paths of polynomials of 119883 belong to1198712(ΩH

119879)

Proposition 38 (Ito formula for polynomials) Let 119883 be aseparable centered Gaussian process with covariance 119877 andassume that 119901 is a polynomial Furthermore assume that for

each polynomial 119901 one has 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) Then for

each 119905 isin [0 119879] one has

119901 (119883119905) = 119901 (119883

0) + int

119905

0

1199011015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11990110158401015840(119883

119904) d119877 (119904 119904)

(63)

if and only if119883sdot1119905belongs to Dom 120575

119879

Remark 39 The message of the above result is that once theprocesses 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) then they automatically

belong to the domain of 120575119879which is a subspace of 1198712(ΩH

119879)

However in order to check 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) one needs

more information on the kernel 119870119879 A sufficient condition is

provided in Corollary 43 which covers many cases of interest

Proof By definition and applying transfer principle we haveto prove that 1199011015840(119883

sdot)1

119905belongs to domain of 120575

119879and that

Eint119905

0

119863119904119866119870

lowast

119879[119901

1015840(119883

sdot) 1

119905] d119904

= E [119866119901 (119883119905)] minus E [119866119901 (119883

0)]

minus1

2int

119905

0

E [11986611990110158401015840(119883

119905)] d119877 (119905 119905)

(64)

for every random variable 119866 from a total subset of 1198712(Ω)In other words it is sufficient to show that (69) is valid forrandom variables of form 119866 = 119868119882

119899(ℎ

otimes119899) where ℎ is a step-

functionNote first that it is sufficient to prove the claim only for

Hermite polynomials119867119896 119896 = 1 Indeed it is well-known

that any polynomial can be expressed as a linear combinationof Hermite polynomials and consequently the result forarbitrary polynomial 119901 follows by linearity

We proceed by induction First it is clear that firsttwo polynomials 119867

0and 119867

1satisfy (64) Furthermore by

assumption 1198671015840

2(119883

sdot)1

119905belongs to Dom 120575

119879from which (64)

is easily deduced by [22 Proposition 133] Assume nextthat the result is valid for Hermite polynomials 119867

119896 119896 =

0 1 119899 Then recall well-known recursion formulas

119867119899+1(119909) = 119909119867

119899(119909) minus 119899119867

119899minus1(119909)

1198671015840

119899(119909) = 119899119867

119899minus1(119909)

(65)

The induction step follows with straightforward calcula-tions by using the recursion formulas above and [22Proposition 133] We leave the details to the reader

We will now illustrate how the result can be generalizedfor functions satisfying the growth condition (61) by usingProposition 38 First note that the growth condition (61) isindeed natural since it guarantees that the left side of (60)is square integrable Consequently since operator 120575

119879is a

mapping from 1198712(ΩH119879) into 1198712(Ω) functions satisfying

(61) are the largest class of functions for which (60) can holdHowever it is not clear in general whether 1198911015840(119883

sdot)1

119905belongs

International Journal of Stochastic Analysis 9

to Dom 120575119879 Indeed for example in [1] the authors posed

additional conditions on the Volterra kernel 119870 to guaranteethis As our main result we show that E1198911015840(119883

sdot)1

1199052

H119879lt infin

implies that (60) holds In other words not only is the Itoformula (60) natural but it is also the only possibility

Theorem 40 (Ito formula for Skorohod integrals) Let 119883 bea separable centered Gaussian process with covariance 119877 suchthat all the polynomials 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) Assume that

119891 isin 1198622 satisfies growth condition (61) and that the variance of

119883 is bounded and of bounded variation If

E100381710038171003817100381710038171198911015840(119883

sdot) 1

119905

10038171003817100381710038171003817

2

H119879lt infin (66)

for any 119905 isin [0 119879] then

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(67)

Proof In this proof we assume for notational simplicity andwith no loss of generality that sup

0le119904le119879119877(119904 119904) = 1

First it is clear that (66) implies that 1198911015840(119883sdot)1

119905belongs to

domain of 120575119879 Hence we only have to prove that

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E [119866119891 (119883119905)] minus E [119866119891 (119883

0)]

minus1

2int

119905

0

E [11986611989110158401015840(119883

119904)] d119877 (119904 119904)

(68)

for every random variable 119866 = 119868119882119899(ℎ

otimes119899)

Now it is well-known that Hermite polynomials whenproperly scaled form an orthogonal system in 1198712(R) whenequipped with the Gaussian measure Now each 119891 satisfyingthe growth condition (61) has a series representation

119891 (119909) =

infin

sum

119896=0

120572119896119867119896(119909) (69)

Indeed the growth condition (61) implies that

intR

100381610038161003816100381610038161198911015840(119909)10038161003816100381610038161003816

2

119890minus1199092(2sup

0le119904le119879119877(119904119904))d119909 lt infin (70)

Furthermore we have

119891 (119883119904) =

infin

sum

119896=0

120572119896119867119896(119883

119904) (71)

where the series converge almost surely and in 1198712(Ω) andsimilar conclusion is valid for derivatives1198911015840(119883

119904) and 11989110158401015840(119883

119904)

Then by applying (66) we obtain that for any 120598 gt 0 thereexists119873 = 119873

120598such that we have

E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598 119899 ge 119873 (72)

where

1198911015840

119899(119883

119904) =

infin

sum

119896=119899

120572119896119867

1015840

119896(119883

119904) (73)

Consequently for random variables of form 119866 = 119868119882119899(ℎ

otimes119899) we

obtain by choosing 119873 large enough and applying Proposi-tion 38 that

E [119866119891 (119883119905)] minus E [119866119891 (119883

0)] minus1

2

sdot int

119905

0

E (11986611989110158401015840(119883

119905)) d119877 (119905 119905)

minus E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598

(74)

Now the left side does not depend on 119899 which concludes theproof

Remark 41 Note that actually it is sufficient to have

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

=

infin

sum

119896=1

120572119896E ⟨119863

119879119866119867

1015840

119896(119883

sdot) 1

119905⟩H119879

(75)

from which the result follows by Proposition 38 Further-more taking account growth condition (61) this is actuallysufficient and necessary condition for formula (60) to holdConsequently our method can also be used to obtain Itoformulas by considering extended domain of 120575

119879(see [9] or

[11]) This is the topic of Section 44

Example 42 It is known that if 119883 = 119861119867 is a fractional Brow-nian motion with119867 gt 14 then 1198911015840(119883

sdot)1

119905satisfies condition

(66) while for 119867 le 14 it does not (see [22 Chapter 5])Consequently a simple application of Theorem 40 coversfractional Brownian motion with 119867 gt 14 For the case119867 le 14 one has to consider extended domain of 120575

119879which is

proved in [9] Consequently in this case we have (75) for any119865 isin S

We end this section by illustrating the power of ourmethod with the following simple corollary which is anextension of [1 Theorem 1]

Corollary 43 Let 119883 be a separable centered continuousGaussian process with covariance 119877 that is bounded such thatthe Fredholm kernel 119870

119879is of bounded variation and

int

119879

0

(int

119879

0

1003817100381710038171003817119883119905minus 119883

119904

10038171003817100381710038171198712(Ω)

10038161003816100381610038161198701198791003816100381610038161003816 (d119905 119904))

2

d119904 lt infin (76)

Then for any 119905 isin [0 119879] one has

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(77)

10 International Journal of Stochastic Analysis

Proof Note that assumption is a Fredholm version of con-dition (K2) in [1] which implies condition (66) Hence theresult follows byTheorem 40

44 Extended Divergence Operator As shown in Section 43the Ito formula (60) is the only possibility However theproblem is that the space 1198712(ΩH

119879) may be too small to

contain the elements 1198911015840(119883sdot)1

119905 In particular it may happen

that not even the process 119883 itself belongs to 1198712(ΩH119879) (see

eg [7] for the case of fractional Brownian motion with119867 le 14) This problem can be overcome by consideringan extended domain of 120575

119879 The idea of extended domain is

to extend the inner product ⟨119906 120593⟩H119879 for simple 120593 to moregeneral processes 119906 and then define extended domain by(47) with a restricted class of test variables 119865 This also givesanother intuitive reason why extended domain of 120575

119879can be

useful indeed here we have proved that Ito formula (60) isthe only possibility and what one essentially needs for suchresult is the following

(i) 119883sdot1119905belongs to Dom 120575

119879

(ii) Equation (75) is valid for functions satisfying (61)

Consequently one should look for extensions of operator 120575119879

such that these two things are satisfiedTo facilitate the extension of domain we make the

following relatively moderate assumption

(H) The function 119905 997891rarr 119877(119905 119904) is of bounded variation on[0 119879] and

sup119905isin[0119879]

int

119879

0

|119877| (d119904 119905) lt infin (78)

Remark 44 Note that we are making the assumption on thecovariance 119877 not the kernel 119870

119879 Hence our case is different

from that of [1] Also [11] assumed absolute continuity in 119877we are satisfied with bounded variation

We will follow the idea from Lei and Nualart [11] andextend the inner product ⟨sdot sdot⟩H119879 beyondH

119879

Consider a step-function 120593Then on the one hand by theisometry property we have

⟨120593 1119905⟩H119879= int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 (79)

where 119892119905(119904) = 119870(119905 119904) isin 119871

2([0 119879]) On the other hand by

using adjoint property (see Remark 20) we obtain

int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 = int

119879

0

120593 (119904) (119870119879119892119905) (d119904) (80)

where computing formally we have

(119870119879119892119905) (d119904) = int

119879

0

119892119905(119906)119870

119879(d119904 119906) d119906

= int

119879

0

119870119879(119905 119906)119870

119879(d119904 119906) d119906 = 119877 (119905 d119904)

(81)

Consequently

⟨120593 1119905⟩H119879= int

119879

0

120593 (119904) 119877 (119905 d119904) (82)

This gives motivation to the following definition similar tothat of [11 Definition 21]

Definition 45 Denote by T119879the space of measurable func-

tions 119892 satisfying

sup119905isin[0119879]

int

119879

0

1003816100381610038161003816119892 (119904)1003816100381610038161003816 |119877| (119905 d119904) lt infin (83)

and let 120593 be a step-function of form 120593 = sum119899119896=11198871198961119905119896 Then we

extend ⟨sdot sdot⟩H119879 toT119879by defining

⟨119892 120593⟩H119879=

119899

sum

119896=1

119887119896int

119879

0

119892 (119904) 119877 (119905119896 d119904) (84)

In particular this implies that for 119892 and 120593 as above we have

⟨1198921119905 120593⟩

H119879= int

119905

0

119892 (119904) d ⟨1119904 120593⟩

H119879 (85)

We define extended domain Dom119864120575119879similarly as in [11]

Definition 46 A process 119906 isin 1198711(ΩT119879) belongs to Dom119864

120575119879

if10038161003816100381610038161003816E ⟨119906119863119865⟩H119879

10038161003816100381610038161003816le 119888

119906119865

2(86)

for any smooth random variable 119865 isin S In this case 120575(119906) isin1198712(Ω) is defined by duality relationship

E [119865120575 (119906)] = E ⟨119906119863119865⟩H119879 (87)

Remark 47 Note that in general Dom 120575119879and Dom119864

120575119879are

not comparable See [11] for discussion

Note now that if a function 119891 satisfies the growthcondition (61) then 1198911015840(119883

sdot)1

119905isin 119871

1(ΩT

119879) since (61) implies

E sup119905isin[0119879]

100381610038161003816100381610038161198911015840(119883

119905)10038161003816100381610038161003816

119901

lt infin (88)

for any 119901 lt (12120582)(sup119905isin[0119879]

119877(119905 119905))minus1 Consequently with

this definition we are able to get rid of the problem thatprocesses might not belong to corresponding H

119879-spaces

Furthermore this implies that the series expansion (69)converges in the norm 1198711(ΩT

119879) defined by

Eint119879

0

|119906 (119904)| |119877| (119905 d119904) (89)

which in turn implies (75) Hence it is straightforward toobtain the following by first showing the result for polynomi-als and then by approximating in a similar manner as done inSection 43 but using the extended domain instead

International Journal of Stochastic Analysis 11

Theorem 48 (Ito formula for extended Skorohod integrals)Let119883 be a separable centered Gaussian process with covariance119877 and assume that 119891 isin 1198622 satisfies growth condition (61)Furthermore assume that (H) holds and that the variance of119883is bounded and is of bounded variationThen for any 119905 isin [0 119879]the process 1198911015840(119883

sdot)1

119905belongs to Dom119864

120575119879and

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(90)

Remark 49 As an application of Theorem 48 it is straight-forward to derive version of Ito-Tanaka formula underadditional conditions which guarantee that for a certainsequence of functions 119891

119899we have the convergence of term

(12) int119905

011989110158401015840

119899(119883

119904)d119877(119904 119904) to the local time For details we refer

to [11] where authors derived such formula under theirassumptions

Finally let us note that the extension to functions 119891(119905 119909)is straightforward where 119891 satisfies the following growthcondition

max [1003816100381610038161003816119891 (119905 119909)1003816100381610038161003816 1003816100381610038161003816120597119905119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119909119891 (119905 119909)

1003816100381610038161003816]

le 119888119890120582|119909|2

(91)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1

Theorem 50 (Ito formula for extended Skorohod integralsII) Let 119883 be a separable centered Gaussian process withcovariance 119877 and assume that 119891 isin 11986212 satisfies growthcondition (91) Furthermore assume that (H) holds and thatthe variance of119883 is bounded and is of bounded variationThenfor any 119905 isin [0 119879] the process 120597

119909119891(sdot 119883

sdot)1

119905belongs to Dom119864

120575119879

and

119891 (119905 119883119905) = 119891 (0 119883

0) + int

119905

0

120597119909119891 (119904 119883

119904) 120575119883

119904

+ int

119905

0

120597119905119891 (119904 119883

119904) d119904

+1

2int

119905

0

120597119909119909119891 (119904 119883

119904) d119877 (119904 119904)

(92)

Proof Taking into account that we have no problems con-cerning processes to belong to the required spaces theformula follows by approximating with polynomials of form119901(119909)119902(119905) and following the proof of Theorem 40

5 Applications

We illustrate how some results transfer easily from theBrownian case to the Gaussian Fredholm processes

51 Equivalence in Law The transfer principle has alreadybeen used in connection with the equivalence of law of

Gaussian processes in for example [24] in the context offractional Brownian motions and in [2] in the context ofGaussianVolterra processes satisfying certain nondegeneracyconditions The following proposition uses the Fredholmrepresentation (8) to give a sufficient condition for theequivalence of general Gaussian processes in terms of theirFredholm kernels

Proposition 51 Let 119883 and be two Gaussian processes withFredholm kernels 119870

119879and

119879 respectively If there exists a

Volterra kernel ℓ isin 1198712([0 119879]2) such that

119879(119905 119904) = 119870

119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 (93)

then119883 and are equivalent in law

Proof Recall that by the Hitsuda representation theorem [5Theorem 63] a centered Gaussian process is equivalent inlaw to a Brownian motion on [0 119879] if and only if there existsa kernel ℓ isin 1198712([0 119879]2) and a Brownian motion119882 such that admits the representation

119905= 119882

119905minus int

119905

0

int

119904

0

ℓ (119904 119906) d119882119906d119904 (94)

Let119883 have the Fredholm representations

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (95)

Then is equivalent to119883 if it admits in law the representa-tion

119905

119889

= int

119879

0

119870119879(119905 119904) d

119904 (96)

where is connected to119882 of (95) by (94)In order to show (96) let

119905= int

119879

0

119879(119905 119904) d1198821015840

119905(97)

be the Fredholm representation of Here 1198821015840 is someBrownian motion Then by using connection (93) and theFubini theorem we obtain

119905= int

119879

0

119879(119905 119904) d1198821015840

119904

119889

= int

119879

0

119879(119905 119904) d119882

119904

= int

119879

0

(119870119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906) d119882

119904

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

4 International Journal of Stochastic Analysis

Remark 13 The Fredholm representation (8) holds also forinfinite intervals that is 119879 = infin if the trace condition (9)holds Unfortunately this is seldom the case

Remark 14 The above proof shows that the Fredholm repre-sentation (8) holds in law However one can also constructthe process119883 via (8) for a given Brownian motion119882 In thiscase representation (8) holds of course in 1198712 Finally notethat in general it is not possible to construct the Brownianmotion in representation (8) from the process 119883 Indeedthere might not be enough randomness in 119883 To construct119882 from119883 one needs that the indicators 1

119905 119905 isin [0 119879] belong

to the range of the operator119870119879

Remark 15 We remark that the separability of 119883 ensuresrepresentation of form (8) where the kernel 119870

119879only satisfies

a weaker condition 119870119879(119905 sdot) isin 119871

2([0 119879]) for all 119905 isin [0 119879]

which may happen if the trace condition (9) fails In thiscase however the associated operator119870

119879does not belong to

1198712([0 119879]) which may be undesirable

Example 16 Let us consider the following very degeneratecase suppose 119883

119905= 119891(119905)120585 where 119891 is deterministic and 120585 is a

standard normal random variable Suppose 119879 gt 1 Then

119883119905= int

119879

0

119891 (119905) 1[01)(119904) d119882s (15)

So 119870119879(119905 119904) = 119891(119905)1

[01)(119904) Now if 119891 isin 1198712([0 119879]) then

condition (9) is satisfied and 119870119879isin 119871

2([0 119879]

2) On the other

hand even if 119891 notin 1198712([0 119879]) we can still write 119883 in form(15) However in this case the kernel 119870

119879does not belong to

1198712([0 119879]

2)

Example 17 Consider a truncated series expansion

119883119905=

119899

sum

119896=1

119890119879

119896(119905) 120585

119896 (16)

where 120585119896are independent standard normal random variables

and

119890119879

119896(119905) = int

119905

0

119879

119896(119904) d119904 (17)

where 119879119896 119896 isin N is an orthonormal basis in 1198712([0 119879]) Now

it is straightforward to check that this process is not purelynondeterministic (see [15] for definition) and consequently119883 cannot have Volterra representation while it is clear that119883 admits a Fredholm representation On the other hand bychoosing the functions 119879

119896to be the trigonometric basis on

1198712([0 119879])119883 is a finite-rank approximation of the Karhunen-

Loeve representation of standard Brownianmotion on [0 119879]Hence by letting 119899 tend to infinity we obtain the standardBrownian motion and hence a Volterra process

Example 18 Let119882 be a standard Brownian motion on [0 119879]and consider the Brownian bridge Now there are two rep-resentations of the Brownian bridge (see [16] and references

therein on the representations of Gaussian bridges) Theorthogonal representation is

119861119905= 119882

119905minus119905

119879119882

119879 (18)

Consequently 119861 has a Fredholm representation with kernel119870119879(119905 119904) = 1

119905(119904) minus 119905119879 The canonical representation of the

Brownian bridge is

119861t = (119879 minus 119905) int119905

0

1

119879 minus 119904d119882

119904 (19)

Consequently the Brownian bridge has also a Volterra-typerepresentation with kernel 119870(119905 119904) = (119879 minus 119905)(119879 minus 119904)

4 Transfer Principle and Stochastic Analysis

41 Wiener Integrals Theorem 22 is the transfer principle inthe context of Wiener integrals The same principle extendsto multiple Wiener integrals and Malliavin calculus later inthe following subsections

Recall that for any kernel Γ isin 1198712([0 119879]2) its associatedoperator on 1198712([0 119879]) is

Γ119891 (119905) = int

119879

0

119891 (119904) Γ (119905 119904) d119904 (20)

Definition 19 (adjoint associated operator) The adjoint asso-ciated operator Γlowast of a kernel Γ isin 1198712([0 119879]2) is defined bylinearly extending the relation

Γlowast1

119905= Γ (119905 sdot) (21)

Remark 20 Thename andnotation of ldquoadjointrdquo for119870lowast

119879comes

from Alos et al [1] where they showed that in their Volterracontext119870lowast

119879admits a kernel and is an adjoint of119870

119879in the sense

that

int

119879

0

119870lowast

119879119891 (119905) 119892 (119905) d119905 = int

119879

0

119891 (119905)119870119879119892 (d119905) (22)

for step-functions 119891 and 119892 belonging to 1198712([0 119879]) It isstraightforward to check that this statement is valid also inour case

Example 21 Suppose the kernel Γ(sdot 119904) is of bounded variationfor all 119904 and that 119891 is nice enough Then

Γlowast119891 (119904) = int

119879

0

119891 (119905) Γ (d119905 119904) (23)

Theorem 22 (transfer principle for Wiener integrals) Let 119883be a separable centered Gaussian process with representation(8) and let 119891 isinH

119879 Then

int

119879

0

119891 (119905) d119883119905= int

119879

0

119870lowast

119879119891 (119905) d119882

119905 (24)

Proof Assume first that 119891 is an elementary function of form

119891 (119905) =

119899

sum

119896=1

1198861198961119860119896

(25)

International Journal of Stochastic Analysis 5

for some disjoint intervals 119860119896= (119905

119896minus1 119905119896] Then the claim

follows by the very definition of the operator119870lowast

119879andWiener

integral with respect to 119883 together with representation(8) Furthermore this shows that 119870lowast

119879provides an isometry

between H119879and 1198712([0 119879]) Hence H

119879can be viewed as a

closure of elementary functions with respect to 119891H119879 =119870

lowast

119879119891

1198712([0119879])

which proves the claim

42 Multiple Wiener Integrals The study of multiple Wienerintegrals goes back to Ito [17] who studied the case ofBrownian motion Later Huang and Cambanis [18] extendedthe notion to general Gaussian processes Dasgupta andKallianpur [19 20] and Perez-Abreu and Tudor [21] studiedmultiple Wiener integrals in the context of fractional Brown-ian motion In [19 20] a method that involved a prior controlmeasure was used and in [21] a transfer principle was usedOur approach here extends the transfer principle methodused in [21]

We begin by recalling multiple Wiener integrals withrespect to Brownian motion and then we apply transferprinciple to generalize the theory to arbitrary Gaussianprocess

Let 119891 be an elementary function on [0 119879]119901 that vanisheson the diagonals that is

119891 =

119899

sum

1198941 119894119901=1

1198861198941 sdotsdotsdot119894119901

1Δ 1198941

timessdotsdotsdottimesΔ 119894119901 (26)

whereΔ119896fl [119905

119896minus1 119905119896) and 119886

1198941 119894119901= 0whenever 119894

119896= 119894

ℓfor some

119896 = ℓ For such 119891 we define the multiple Wiener integral as

119868119882

119879119901(119891) fl int

119879

0

sdot sdot sdot int

119879

0

119891 (1199051 119905

119901) 120575119882

1199051sdot sdot sdot 120575119882

119905119901

fl119899

sum

1198941i119901=11198861198941 119894119901Δ119882

1199051sdot sdot sdot Δ119882

119905119901

(27)

where we have denoted Δ119882119905119896fl119882

119905119896minus119882

119905119896minus1 For 119901 = 0 we set

119868119882

0(119891) = 119891 Now it can be shown that elementary functions

that vanish on the diagonals are dense in 1198712([0 119879]119901)Thusone can extend the operator 119868119882

119879119901to the space 1198712([0 119879]119901) This

extension is called the multiple Wiener integral with respectto the Brownian motion

Remark 23 It is well-known that 119868119882119879119901(119891) can be under-

stood as multiple or iterated Ito integrals if and only if119891(119905

1 119905

119901) = 0 unless 119905

1le sdot sdot sdot le 119905

119901 In this case we have

119868119882

119879119901(119891)

= 119901 int

119879

0

int

119905119901

0

sdot sdot sdot int

1199052

0

119891 (1199051 119905

119901) d119882

1199051sdot sdot sdot d119882

119905119901

(28)

For the case of Gaussian processes that are not martingalesthis fact is totally useless

For a general Gaussian process119883 recall first the Hermitepolynomials

119867119901(119909) fl

(minus1)119901

119901119890(12)119909

2 d119901

d119909119901(119890

minus(12)1199092

) (29)

For any 119901 ge 1 let the 119901th Wiener chaos of 119883 be theclosed linear subspace of 1198712(Ω) generated by the randomvariables 119867

119901(119883(120593)) 120593 isin H 120593H = 1 where 119867119901

is the119901th Hermite polynomial It is well-known that the mapping119868119883

119901(120593

otimes119901) = 119901119867

119901(119883(120593)) provides a linear isometry between

the symmetric tensor product H⊙119901 and the 119901th Wienerchaos The random variables 119868119883

119901(120593

otimes119901) are called multiple

Wiener integrals of order 119901 with respect to the Gaussianprocess119883

Let us now consider the multipleWiener integrals 119868119879119901

fora general Gaussian process119883 We define the multiple integral119868119879119901

by using the transfer principle in Definition 25 and laterargue that this is the ldquocorrectrdquo way of defining them So let119883 be a centered Gaussian process on [0 119879]with covariance 119877and representation (8) with kernel 119870

119879

Definition 24 (119901-fold adjoint associated operator) Let119870119879be

the kernel in (8) and let119870lowast

119879be its adjoint associated operator

Define

119870lowast

119879119901fl (119870lowast

119879)otimes119901

(30)

In the same way define

H119879119901

fl Hotimes119901

119879

H0

119879119901fl (Hotimes119901

119879)0

(31)

Here the tensor products are understood in the sense ofHilbert spaces that is they are closed under the inner productcorresponding to the 119901-fold product of the underlying innerproduct

Definition 25 Let 119883 be a centered Gaussian process withrepresentation (8) and let 119891 isinH

119879119901 Then

119868119879119901(119891) fl 119868119882

119879119901(119870

lowast

119879119901119891) (32)

The following example should convince the reader thatthis is indeed the correct definition

Example 26 Let 119901 = 2 and let ℎ = ℎ1otimesℎ

2 where both ℎ

1and

ℎ2are step-functions Then

(119870lowast

1198792ℎ) (119909 119910) = (119870

lowast

119879ℎ1) (119909) (119870

lowast

119879ℎ2) (119910)

119868119882

1198792(119870

lowast

1198792119891)

= int

119879

0

119870lowast

119879ℎ1(V) d119882V

sdot int

119879

0

119870lowast

119879ℎ2(119906) d119882

119906minus ⟨119870

lowast

119879ℎ1 119870

lowast

119879ℎ2⟩1198712([0119879])

= 119883 (ℎ1)119883 (ℎ

2) minus ⟨ℎ

1 ℎ

2⟩H119879

(33)

6 International Journal of Stochastic Analysis

The following proposition shows that our approach todefine multiple Wiener integrals is consistent with the tra-ditional approach where multiple Wiener integrals for moregeneral Gaussian process 119883 are defined as the closed linearspace generated by Hermite polynomials

Proposition 27 Let 119867119901be the 119901th Hermite polynomial and

let ℎ isinH119879 Then

119868119879119901(ℎ

otimes119901) = 119901 ℎ

119901

H119879119867119901(119883 (ℎ)

ℎH119879

) (34)

Proof First note that without loss of generalitywe can assumeℎH119879

= 1Now by the definition of multipleWiener integralwith respect to119883 we have

119868119879119901(ℎ

otimes119901) = 119868

119882

119879119901(119870

lowast

119879119901ℎotimes119901) (35)

where

119870lowast

119879119901ℎotimes119901= (119870

lowast

119879ℎ)

otimes119901

(36)

Consequently by [22 Proposition 114] we obtain

119868119879119901(ℎ

otimes119901) = 119901119867

119901(119882 (119870

lowast

119879ℎ)) (37)

which implies the result together withTheorem 22

Proposition 27 extends to the following product formulawhich is also well-known in the Gaussianmartingale case butapparently new for general Gaussian processes Again theproof is straightforward application of transfer principle

Proposition 28 Let 119891 isinH119879119901

and 119892 isinH119879119902Then

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimes

119870119879119903119892)

(38)

where

119891otimes119870119879119903119892 = (119870

lowast

119879119901+119902minus2119903)minus1

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (39)

and (119870lowast

119879119901+119902minus2119903)minus1 denotes the preimage of 119870lowast

119879119901+119902minus2119903

Proof The proof follows directly from the definition of119868119879119901(119891) and [22 Proposition 113]

Example 29 Let 119891 isin H119879119901

and 119892 isin H119879119902

be of forms119891(119909

1 119909

119901) = prod

119901

119896=1119891119896(119909

119896) and119892(119910

1 119910

119901) = prod

119902

119896=1119892119896(119910

119896)

Then

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (119909

1 119909

119901minus119903 119910

1 119910

119902minus119903)

= int[0119879]119903

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119903

prod

119895=1

119870lowast

119879119891119901minus119903+119895

(119904119895)

sdot

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot

119903

prod

119895=1

119870lowast

119879119892119902minus119903+119895(119904

119895) d119904

1sdot sdot sdot d119904

119903

=

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot ⟨

119903

prod

119895=1

119891119901minus119903+119895

119903

prod

119895=1

119892119902minus119903+119895⟩

H119879119903

(40)

Hence119891otimes

119870119879119903119892

=

119901minus119903

prod

119896=1

119891119896(119909

119896)

119902minus119903

prod

119896=1

119892119896(119910

119896)⟨

119903

prod

119895=1

119891119895

119903

prod

119895=1

119892119895⟩

H119879119903

(41)

Remark 30 In the literature multiple Wiener integrals areusually defined as the closed linear space spanned byHermitepolynomials In such a case Proposition 27 is clearly true bythe very definition Furthermore one has a multiplicationformula (see eg [23])

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimesH119879119903

119892)

(42)

where 119891otimesH119879119903119892 denotes symmetrization of tensor product

119891otimesH119879119903119892 =

infin

sum

1198941 119894119903=1

⟨119891 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

otimes ⟨119892 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

(43)

and 119890119896 119896 = 1 is a complete orthonormal basis of the

Hilbert space H119879 Clearly by Proposition 27 both formulas

coincide This also shows that (39) is well-defined

43 Malliavin Calculus and Skorohod Integrals We begin byrecalling some basic facts on Malliavin calculus

Definition 31 Denote by S the space of all smooth randomvariables of the form

119865 = 119891 (119883 (1205931) 119883 (120593

119899)) 120593

1 120593

119899isinH

119879 (44)

International Journal of Stochastic Analysis 7

where 119891 isin 119862infin119887(R119899) that is 119891 and all its derivatives are

bounded The Malliavin derivative 119863119879119865 = 119863

119883

119879119865 of 119865 is an

element of 1198712(ΩH119879) defined by

119863119879119865 =

119899

sum

119894=1

120597119894119891 (119883 (120593

1) 119883 (120593

119899)) 120593

119894 (45)

In particular119863119879119883119905= 1

119905

Definition 32 Let D12= D12

119883be the Hilbert space of all

square integrable Malliavin differentiable random variablesdefined as the closure of S with respect to norm

1198652

12= E [|119865|

2] + E [

10038171003817100381710038171198631198791198651003817100381710038171003817

2

H119879] (46)

The divergence operator 120575119879is defined as the adjoint

operator of the Malliavin derivative119863119879

Definition 33 The domain Dom 120575119879of the operator 120575

119879is the

set of random variables 119906 isin 1198712(ΩH119879) satisfying

10038161003816100381610038161003816E ⟨119863

119879119865 119906⟩

H119879

10038161003816100381610038161003816le 119888

119906119865

1198712 (47)

for any 119865 isin D12 and some constant 119888119906depending only on 119906

For 119906 isin Dom 120575119879the divergence operator 120575

119879(119906) is a square

integrable random variable defined by the duality relation

E [119865120575119879(119906)] = E ⟨119863

119879119865 119906⟩

H119879(48)

for all 119865 isin D12

Remark 34 It is well-known that D12sub Dom 120575

119879

We use the notation

120575119879(119906) = int

119879

0

119906119904120575119883

119904 (49)

Theorem 35 (transfer principle for Malliavin calculus) Let119883 be a separable centered Gaussian process with Fredholmrepresentation (8) Let 119863

119879and 120575

119879be the Malliavin derivative

and the Skorohod integral with respect to119883 on [0 119879] Similarlylet 119863119882

119879and 120575119882

119879be the Malliavin derivative and the Skorohod

integral with respect to the Brownianmotion119882 of (8) restrictedon [0 119879] Then

120575119879= 120575

119882

119879119870lowast

119879

119870lowast

119879119863119879= 119863

119882

119879

(50)

Proof The proof follows directly from transfer principle andthe isometry provided by 119870lowast

119879with the same arguments as in

[1] Indeed by isometry we have

H119879= (119870

lowast

119879)minus1

(1198712([0 119879])) (51)

where (119870lowast

119879)minus1 denotes the preimage which implies that

D12(H

119879) = (119870

lowast

119879)minus1

(D12

119882(119871

2([0 119879]))) (52)

which justifies 119870lowast

119879119863119879= 119863

119882

119879 Furthermore we have relation

E ⟨119906119863119879119865⟩

H119879= E ⟨119870

lowast

119879119906119863

119882

119879119865⟩

1198712([0119879])

(53)

for any smooth random variable 119865 and 119906 isin 1198712(ΩH119879)

Hence by the very definition ofDom 120575 and transfer principlewe obtain

Dom 120575119879= (119870

lowast

119879)minus1

(Dom 120575119882119879) (54)

and 120575119879(119906) = 120575

119882

119879(119870

lowast

119879119906) proving the claim

Now we are ready to show that the definition of themultiple Wiener integral 119868

119879119901in Section 42 is correct in the

sense that it agrees with the iterated Skorohod integral

Proposition 36 Let ℎ isin H119879119901

be of form ℎ(1199091 119909

119901) =

prod119901

119896=1ℎ119896(119909

119896) Then ℎ is iteratively 119901 times Skorohod integrable

and

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901) 120575119883

1199051sdot sdot sdot 120575119883

119905119901= 119868

119879119901(ℎ) (55)

Moreover if ℎ isin H0

119879119901is such that it is 119901 times iteratively

Skorohod integrable then (55) still holds

Proof Again the idea is to use the transfer principle togetherwith induction Note first that the statement is true for 119901 = 1by definition and assume next that the statement is validfor 119896 = 1 119901 We denote 119891

119895= prod

119895

119896=1ℎ119896(119909

119896) Hence by

induction assumption we have

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901 119905V) 1205751198831199051

sdot sdot sdot 120575119883119905119901120575119883V

= int

119879

0

119868119879119901(119891

119901) ℎ

119901+1(V) 120575119883V

(56)

Put now 119865 = 119868119879119901(119891

119901) and 119906(119905) = ℎ

119901+1(119905) Hence by [22

Proposition 133] and by applying the transfer principle weobtain that 119865119906 belongs to Dom 120575

119879and

120575119879(119865119906) = 120575

119879(119906) 119865 minus ⟨119863

119905119865 119906 (119905)⟩

H119879

= 119868119882

119879(119870

lowast

119879ℎ119901+1) 119868

119882

119879119901(119870

lowast

119879119901119891119901)

minus 119901 ⟨119868119879119901minus1(119891

119901(sdot 119905)) ℎ

119901+1(119905)⟩

H119879

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119882

119879119901minus1(119870

lowast

119879119901minus1119891119901otimes1119870lowast

119879ℎ119901+1)

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119879119901minus1(119891

119901otimes1198701198791ℎ119901+1)

(57)

Hence the result is valid also for 119901+ 1 by Proposition 28 with119902 = 1

The claim for general ℎ isinH0

119879119901follows by approximating

with a product of simple function

8 International Journal of Stochastic Analysis

Remark 37 Note that (55) does not hold for arbitrary ℎ isinH0

119879119901in general without the a priori assumption of 119901 times

iterative Skorohod integrability For example let 119901 = 2 and119883 = 119861

119867 be a fractional Brownian motion with119867 le 14 anddefine ℎ

119905(119904 V) = 1

119905(119904)1

119904(V) for some fixed 119905 isin [0 119879] Then

int

119879

0

ℎ119905(119904 V) 120575119883V = 119883119904

1119905(119904) (58)

But119883sdot1119905does not belong to Dom 120575

119879(see [7])

We end this section by providing an extension of Ito for-mulas provided by Alos et al [1] They considered GaussianVolterra processes that is they assumed the representation

119883119905= int

119905

0

119870 (119905 119904) d119882119904 (59)

where the kernel119870 satisfied certain technical assumptions In[1] it was proved that in the case of Volterra processes one has

119891 (119883119905) = 119891 (0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(60)

if 119891 satisfies the growth condition

max [1003816100381610038161003816119891 (119909)1003816100381610038161003816 100381610038161003816100381610038161198911015840(119909)100381610038161003816100381610038161003816100381610038161003816100381611989110158401015840(119909)10038161003816100381610038161003816] le 119888119890

120582|119909|2

(61)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1 In the

following we will consider different approach which enablesus to

(i) prove that such formula holds with minimal require-ments

(ii) give more instructive proof of such result(iii) extend the result from Volterra context to more

general Gaussian processes(iv) drop some technical assumptions posed in [1]

For simplicity we assume that the variance of 119883 is ofbounded variation to guarantee the existence of the integral

int

119879

0

11989110158401015840(119883

119905) d119877 (119905 119905) (62)

If the variance is not of bounded variation then integral (62)may be understood by integration by parts if 11989110158401015840 is smoothenough or in the general case via the inner product ⟨sdot sdot⟩H119879 InTheorem 40 we also have to assume that the variance of119883is bounded

The result for polynomials is straightforward once weassume that the paths of polynomials of 119883 belong to1198712(ΩH

119879)

Proposition 38 (Ito formula for polynomials) Let 119883 be aseparable centered Gaussian process with covariance 119877 andassume that 119901 is a polynomial Furthermore assume that for

each polynomial 119901 one has 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) Then for

each 119905 isin [0 119879] one has

119901 (119883119905) = 119901 (119883

0) + int

119905

0

1199011015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11990110158401015840(119883

119904) d119877 (119904 119904)

(63)

if and only if119883sdot1119905belongs to Dom 120575

119879

Remark 39 The message of the above result is that once theprocesses 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) then they automatically

belong to the domain of 120575119879which is a subspace of 1198712(ΩH

119879)

However in order to check 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) one needs

more information on the kernel 119870119879 A sufficient condition is

provided in Corollary 43 which covers many cases of interest

Proof By definition and applying transfer principle we haveto prove that 1199011015840(119883

sdot)1

119905belongs to domain of 120575

119879and that

Eint119905

0

119863119904119866119870

lowast

119879[119901

1015840(119883

sdot) 1

119905] d119904

= E [119866119901 (119883119905)] minus E [119866119901 (119883

0)]

minus1

2int

119905

0

E [11986611990110158401015840(119883

119905)] d119877 (119905 119905)

(64)

for every random variable 119866 from a total subset of 1198712(Ω)In other words it is sufficient to show that (69) is valid forrandom variables of form 119866 = 119868119882

119899(ℎ

otimes119899) where ℎ is a step-

functionNote first that it is sufficient to prove the claim only for

Hermite polynomials119867119896 119896 = 1 Indeed it is well-known

that any polynomial can be expressed as a linear combinationof Hermite polynomials and consequently the result forarbitrary polynomial 119901 follows by linearity

We proceed by induction First it is clear that firsttwo polynomials 119867

0and 119867

1satisfy (64) Furthermore by

assumption 1198671015840

2(119883

sdot)1

119905belongs to Dom 120575

119879from which (64)

is easily deduced by [22 Proposition 133] Assume nextthat the result is valid for Hermite polynomials 119867

119896 119896 =

0 1 119899 Then recall well-known recursion formulas

119867119899+1(119909) = 119909119867

119899(119909) minus 119899119867

119899minus1(119909)

1198671015840

119899(119909) = 119899119867

119899minus1(119909)

(65)

The induction step follows with straightforward calcula-tions by using the recursion formulas above and [22Proposition 133] We leave the details to the reader

We will now illustrate how the result can be generalizedfor functions satisfying the growth condition (61) by usingProposition 38 First note that the growth condition (61) isindeed natural since it guarantees that the left side of (60)is square integrable Consequently since operator 120575

119879is a

mapping from 1198712(ΩH119879) into 1198712(Ω) functions satisfying

(61) are the largest class of functions for which (60) can holdHowever it is not clear in general whether 1198911015840(119883

sdot)1

119905belongs

International Journal of Stochastic Analysis 9

to Dom 120575119879 Indeed for example in [1] the authors posed

additional conditions on the Volterra kernel 119870 to guaranteethis As our main result we show that E1198911015840(119883

sdot)1

1199052

H119879lt infin

implies that (60) holds In other words not only is the Itoformula (60) natural but it is also the only possibility

Theorem 40 (Ito formula for Skorohod integrals) Let 119883 bea separable centered Gaussian process with covariance 119877 suchthat all the polynomials 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) Assume that

119891 isin 1198622 satisfies growth condition (61) and that the variance of

119883 is bounded and of bounded variation If

E100381710038171003817100381710038171198911015840(119883

sdot) 1

119905

10038171003817100381710038171003817

2

H119879lt infin (66)

for any 119905 isin [0 119879] then

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(67)

Proof In this proof we assume for notational simplicity andwith no loss of generality that sup

0le119904le119879119877(119904 119904) = 1

First it is clear that (66) implies that 1198911015840(119883sdot)1

119905belongs to

domain of 120575119879 Hence we only have to prove that

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E [119866119891 (119883119905)] minus E [119866119891 (119883

0)]

minus1

2int

119905

0

E [11986611989110158401015840(119883

119904)] d119877 (119904 119904)

(68)

for every random variable 119866 = 119868119882119899(ℎ

otimes119899)

Now it is well-known that Hermite polynomials whenproperly scaled form an orthogonal system in 1198712(R) whenequipped with the Gaussian measure Now each 119891 satisfyingthe growth condition (61) has a series representation

119891 (119909) =

infin

sum

119896=0

120572119896119867119896(119909) (69)

Indeed the growth condition (61) implies that

intR

100381610038161003816100381610038161198911015840(119909)10038161003816100381610038161003816

2

119890minus1199092(2sup

0le119904le119879119877(119904119904))d119909 lt infin (70)

Furthermore we have

119891 (119883119904) =

infin

sum

119896=0

120572119896119867119896(119883

119904) (71)

where the series converge almost surely and in 1198712(Ω) andsimilar conclusion is valid for derivatives1198911015840(119883

119904) and 11989110158401015840(119883

119904)

Then by applying (66) we obtain that for any 120598 gt 0 thereexists119873 = 119873

120598such that we have

E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598 119899 ge 119873 (72)

where

1198911015840

119899(119883

119904) =

infin

sum

119896=119899

120572119896119867

1015840

119896(119883

119904) (73)

Consequently for random variables of form 119866 = 119868119882119899(ℎ

otimes119899) we

obtain by choosing 119873 large enough and applying Proposi-tion 38 that

E [119866119891 (119883119905)] minus E [119866119891 (119883

0)] minus1

2

sdot int

119905

0

E (11986611989110158401015840(119883

119905)) d119877 (119905 119905)

minus E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598

(74)

Now the left side does not depend on 119899 which concludes theproof

Remark 41 Note that actually it is sufficient to have

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

=

infin

sum

119896=1

120572119896E ⟨119863

119879119866119867

1015840

119896(119883

sdot) 1

119905⟩H119879

(75)

from which the result follows by Proposition 38 Further-more taking account growth condition (61) this is actuallysufficient and necessary condition for formula (60) to holdConsequently our method can also be used to obtain Itoformulas by considering extended domain of 120575

119879(see [9] or

[11]) This is the topic of Section 44

Example 42 It is known that if 119883 = 119861119867 is a fractional Brow-nian motion with119867 gt 14 then 1198911015840(119883

sdot)1

119905satisfies condition

(66) while for 119867 le 14 it does not (see [22 Chapter 5])Consequently a simple application of Theorem 40 coversfractional Brownian motion with 119867 gt 14 For the case119867 le 14 one has to consider extended domain of 120575

119879which is

proved in [9] Consequently in this case we have (75) for any119865 isin S

We end this section by illustrating the power of ourmethod with the following simple corollary which is anextension of [1 Theorem 1]

Corollary 43 Let 119883 be a separable centered continuousGaussian process with covariance 119877 that is bounded such thatthe Fredholm kernel 119870

119879is of bounded variation and

int

119879

0

(int

119879

0

1003817100381710038171003817119883119905minus 119883

119904

10038171003817100381710038171198712(Ω)

10038161003816100381610038161198701198791003816100381610038161003816 (d119905 119904))

2

d119904 lt infin (76)

Then for any 119905 isin [0 119879] one has

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(77)

10 International Journal of Stochastic Analysis

Proof Note that assumption is a Fredholm version of con-dition (K2) in [1] which implies condition (66) Hence theresult follows byTheorem 40

44 Extended Divergence Operator As shown in Section 43the Ito formula (60) is the only possibility However theproblem is that the space 1198712(ΩH

119879) may be too small to

contain the elements 1198911015840(119883sdot)1

119905 In particular it may happen

that not even the process 119883 itself belongs to 1198712(ΩH119879) (see

eg [7] for the case of fractional Brownian motion with119867 le 14) This problem can be overcome by consideringan extended domain of 120575

119879 The idea of extended domain is

to extend the inner product ⟨119906 120593⟩H119879 for simple 120593 to moregeneral processes 119906 and then define extended domain by(47) with a restricted class of test variables 119865 This also givesanother intuitive reason why extended domain of 120575

119879can be

useful indeed here we have proved that Ito formula (60) isthe only possibility and what one essentially needs for suchresult is the following

(i) 119883sdot1119905belongs to Dom 120575

119879

(ii) Equation (75) is valid for functions satisfying (61)

Consequently one should look for extensions of operator 120575119879

such that these two things are satisfiedTo facilitate the extension of domain we make the

following relatively moderate assumption

(H) The function 119905 997891rarr 119877(119905 119904) is of bounded variation on[0 119879] and

sup119905isin[0119879]

int

119879

0

|119877| (d119904 119905) lt infin (78)

Remark 44 Note that we are making the assumption on thecovariance 119877 not the kernel 119870

119879 Hence our case is different

from that of [1] Also [11] assumed absolute continuity in 119877we are satisfied with bounded variation

We will follow the idea from Lei and Nualart [11] andextend the inner product ⟨sdot sdot⟩H119879 beyondH

119879

Consider a step-function 120593Then on the one hand by theisometry property we have

⟨120593 1119905⟩H119879= int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 (79)

where 119892119905(119904) = 119870(119905 119904) isin 119871

2([0 119879]) On the other hand by

using adjoint property (see Remark 20) we obtain

int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 = int

119879

0

120593 (119904) (119870119879119892119905) (d119904) (80)

where computing formally we have

(119870119879119892119905) (d119904) = int

119879

0

119892119905(119906)119870

119879(d119904 119906) d119906

= int

119879

0

119870119879(119905 119906)119870

119879(d119904 119906) d119906 = 119877 (119905 d119904)

(81)

Consequently

⟨120593 1119905⟩H119879= int

119879

0

120593 (119904) 119877 (119905 d119904) (82)

This gives motivation to the following definition similar tothat of [11 Definition 21]

Definition 45 Denote by T119879the space of measurable func-

tions 119892 satisfying

sup119905isin[0119879]

int

119879

0

1003816100381610038161003816119892 (119904)1003816100381610038161003816 |119877| (119905 d119904) lt infin (83)

and let 120593 be a step-function of form 120593 = sum119899119896=11198871198961119905119896 Then we

extend ⟨sdot sdot⟩H119879 toT119879by defining

⟨119892 120593⟩H119879=

119899

sum

119896=1

119887119896int

119879

0

119892 (119904) 119877 (119905119896 d119904) (84)

In particular this implies that for 119892 and 120593 as above we have

⟨1198921119905 120593⟩

H119879= int

119905

0

119892 (119904) d ⟨1119904 120593⟩

H119879 (85)

We define extended domain Dom119864120575119879similarly as in [11]

Definition 46 A process 119906 isin 1198711(ΩT119879) belongs to Dom119864

120575119879

if10038161003816100381610038161003816E ⟨119906119863119865⟩H119879

10038161003816100381610038161003816le 119888

119906119865

2(86)

for any smooth random variable 119865 isin S In this case 120575(119906) isin1198712(Ω) is defined by duality relationship

E [119865120575 (119906)] = E ⟨119906119863119865⟩H119879 (87)

Remark 47 Note that in general Dom 120575119879and Dom119864

120575119879are

not comparable See [11] for discussion

Note now that if a function 119891 satisfies the growthcondition (61) then 1198911015840(119883

sdot)1

119905isin 119871

1(ΩT

119879) since (61) implies

E sup119905isin[0119879]

100381610038161003816100381610038161198911015840(119883

119905)10038161003816100381610038161003816

119901

lt infin (88)

for any 119901 lt (12120582)(sup119905isin[0119879]

119877(119905 119905))minus1 Consequently with

this definition we are able to get rid of the problem thatprocesses might not belong to corresponding H

119879-spaces

Furthermore this implies that the series expansion (69)converges in the norm 1198711(ΩT

119879) defined by

Eint119879

0

|119906 (119904)| |119877| (119905 d119904) (89)

which in turn implies (75) Hence it is straightforward toobtain the following by first showing the result for polynomi-als and then by approximating in a similar manner as done inSection 43 but using the extended domain instead

International Journal of Stochastic Analysis 11

Theorem 48 (Ito formula for extended Skorohod integrals)Let119883 be a separable centered Gaussian process with covariance119877 and assume that 119891 isin 1198622 satisfies growth condition (61)Furthermore assume that (H) holds and that the variance of119883is bounded and is of bounded variationThen for any 119905 isin [0 119879]the process 1198911015840(119883

sdot)1

119905belongs to Dom119864

120575119879and

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(90)

Remark 49 As an application of Theorem 48 it is straight-forward to derive version of Ito-Tanaka formula underadditional conditions which guarantee that for a certainsequence of functions 119891

119899we have the convergence of term

(12) int119905

011989110158401015840

119899(119883

119904)d119877(119904 119904) to the local time For details we refer

to [11] where authors derived such formula under theirassumptions

Finally let us note that the extension to functions 119891(119905 119909)is straightforward where 119891 satisfies the following growthcondition

max [1003816100381610038161003816119891 (119905 119909)1003816100381610038161003816 1003816100381610038161003816120597119905119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119909119891 (119905 119909)

1003816100381610038161003816]

le 119888119890120582|119909|2

(91)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1

Theorem 50 (Ito formula for extended Skorohod integralsII) Let 119883 be a separable centered Gaussian process withcovariance 119877 and assume that 119891 isin 11986212 satisfies growthcondition (91) Furthermore assume that (H) holds and thatthe variance of119883 is bounded and is of bounded variationThenfor any 119905 isin [0 119879] the process 120597

119909119891(sdot 119883

sdot)1

119905belongs to Dom119864

120575119879

and

119891 (119905 119883119905) = 119891 (0 119883

0) + int

119905

0

120597119909119891 (119904 119883

119904) 120575119883

119904

+ int

119905

0

120597119905119891 (119904 119883

119904) d119904

+1

2int

119905

0

120597119909119909119891 (119904 119883

119904) d119877 (119904 119904)

(92)

Proof Taking into account that we have no problems con-cerning processes to belong to the required spaces theformula follows by approximating with polynomials of form119901(119909)119902(119905) and following the proof of Theorem 40

5 Applications

We illustrate how some results transfer easily from theBrownian case to the Gaussian Fredholm processes

51 Equivalence in Law The transfer principle has alreadybeen used in connection with the equivalence of law of

Gaussian processes in for example [24] in the context offractional Brownian motions and in [2] in the context ofGaussianVolterra processes satisfying certain nondegeneracyconditions The following proposition uses the Fredholmrepresentation (8) to give a sufficient condition for theequivalence of general Gaussian processes in terms of theirFredholm kernels

Proposition 51 Let 119883 and be two Gaussian processes withFredholm kernels 119870

119879and

119879 respectively If there exists a

Volterra kernel ℓ isin 1198712([0 119879]2) such that

119879(119905 119904) = 119870

119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 (93)

then119883 and are equivalent in law

Proof Recall that by the Hitsuda representation theorem [5Theorem 63] a centered Gaussian process is equivalent inlaw to a Brownian motion on [0 119879] if and only if there existsa kernel ℓ isin 1198712([0 119879]2) and a Brownian motion119882 such that admits the representation

119905= 119882

119905minus int

119905

0

int

119904

0

ℓ (119904 119906) d119882119906d119904 (94)

Let119883 have the Fredholm representations

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (95)

Then is equivalent to119883 if it admits in law the representa-tion

119905

119889

= int

119879

0

119870119879(119905 119904) d

119904 (96)

where is connected to119882 of (95) by (94)In order to show (96) let

119905= int

119879

0

119879(119905 119904) d1198821015840

119905(97)

be the Fredholm representation of Here 1198821015840 is someBrownian motion Then by using connection (93) and theFubini theorem we obtain

119905= int

119879

0

119879(119905 119904) d1198821015840

119904

119889

= int

119879

0

119879(119905 119904) d119882

119904

= int

119879

0

(119870119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906) d119882

119904

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

International Journal of Stochastic Analysis 5

for some disjoint intervals 119860119896= (119905

119896minus1 119905119896] Then the claim

follows by the very definition of the operator119870lowast

119879andWiener

integral with respect to 119883 together with representation(8) Furthermore this shows that 119870lowast

119879provides an isometry

between H119879and 1198712([0 119879]) Hence H

119879can be viewed as a

closure of elementary functions with respect to 119891H119879 =119870

lowast

119879119891

1198712([0119879])

which proves the claim

42 Multiple Wiener Integrals The study of multiple Wienerintegrals goes back to Ito [17] who studied the case ofBrownian motion Later Huang and Cambanis [18] extendedthe notion to general Gaussian processes Dasgupta andKallianpur [19 20] and Perez-Abreu and Tudor [21] studiedmultiple Wiener integrals in the context of fractional Brown-ian motion In [19 20] a method that involved a prior controlmeasure was used and in [21] a transfer principle was usedOur approach here extends the transfer principle methodused in [21]

We begin by recalling multiple Wiener integrals withrespect to Brownian motion and then we apply transferprinciple to generalize the theory to arbitrary Gaussianprocess

Let 119891 be an elementary function on [0 119879]119901 that vanisheson the diagonals that is

119891 =

119899

sum

1198941 119894119901=1

1198861198941 sdotsdotsdot119894119901

1Δ 1198941

timessdotsdotsdottimesΔ 119894119901 (26)

whereΔ119896fl [119905

119896minus1 119905119896) and 119886

1198941 119894119901= 0whenever 119894

119896= 119894

ℓfor some

119896 = ℓ For such 119891 we define the multiple Wiener integral as

119868119882

119879119901(119891) fl int

119879

0

sdot sdot sdot int

119879

0

119891 (1199051 119905

119901) 120575119882

1199051sdot sdot sdot 120575119882

119905119901

fl119899

sum

1198941i119901=11198861198941 119894119901Δ119882

1199051sdot sdot sdot Δ119882

119905119901

(27)

where we have denoted Δ119882119905119896fl119882

119905119896minus119882

119905119896minus1 For 119901 = 0 we set

119868119882

0(119891) = 119891 Now it can be shown that elementary functions

that vanish on the diagonals are dense in 1198712([0 119879]119901)Thusone can extend the operator 119868119882

119879119901to the space 1198712([0 119879]119901) This

extension is called the multiple Wiener integral with respectto the Brownian motion

Remark 23 It is well-known that 119868119882119879119901(119891) can be under-

stood as multiple or iterated Ito integrals if and only if119891(119905

1 119905

119901) = 0 unless 119905

1le sdot sdot sdot le 119905

119901 In this case we have

119868119882

119879119901(119891)

= 119901 int

119879

0

int

119905119901

0

sdot sdot sdot int

1199052

0

119891 (1199051 119905

119901) d119882

1199051sdot sdot sdot d119882

119905119901

(28)

For the case of Gaussian processes that are not martingalesthis fact is totally useless

For a general Gaussian process119883 recall first the Hermitepolynomials

119867119901(119909) fl

(minus1)119901

119901119890(12)119909

2 d119901

d119909119901(119890

minus(12)1199092

) (29)

For any 119901 ge 1 let the 119901th Wiener chaos of 119883 be theclosed linear subspace of 1198712(Ω) generated by the randomvariables 119867

119901(119883(120593)) 120593 isin H 120593H = 1 where 119867119901

is the119901th Hermite polynomial It is well-known that the mapping119868119883

119901(120593

otimes119901) = 119901119867

119901(119883(120593)) provides a linear isometry between

the symmetric tensor product H⊙119901 and the 119901th Wienerchaos The random variables 119868119883

119901(120593

otimes119901) are called multiple

Wiener integrals of order 119901 with respect to the Gaussianprocess119883

Let us now consider the multipleWiener integrals 119868119879119901

fora general Gaussian process119883 We define the multiple integral119868119879119901

by using the transfer principle in Definition 25 and laterargue that this is the ldquocorrectrdquo way of defining them So let119883 be a centered Gaussian process on [0 119879]with covariance 119877and representation (8) with kernel 119870

119879

Definition 24 (119901-fold adjoint associated operator) Let119870119879be

the kernel in (8) and let119870lowast

119879be its adjoint associated operator

Define

119870lowast

119879119901fl (119870lowast

119879)otimes119901

(30)

In the same way define

H119879119901

fl Hotimes119901

119879

H0

119879119901fl (Hotimes119901

119879)0

(31)

Here the tensor products are understood in the sense ofHilbert spaces that is they are closed under the inner productcorresponding to the 119901-fold product of the underlying innerproduct

Definition 25 Let 119883 be a centered Gaussian process withrepresentation (8) and let 119891 isinH

119879119901 Then

119868119879119901(119891) fl 119868119882

119879119901(119870

lowast

119879119901119891) (32)

The following example should convince the reader thatthis is indeed the correct definition

Example 26 Let 119901 = 2 and let ℎ = ℎ1otimesℎ

2 where both ℎ

1and

ℎ2are step-functions Then

(119870lowast

1198792ℎ) (119909 119910) = (119870

lowast

119879ℎ1) (119909) (119870

lowast

119879ℎ2) (119910)

119868119882

1198792(119870

lowast

1198792119891)

= int

119879

0

119870lowast

119879ℎ1(V) d119882V

sdot int

119879

0

119870lowast

119879ℎ2(119906) d119882

119906minus ⟨119870

lowast

119879ℎ1 119870

lowast

119879ℎ2⟩1198712([0119879])

= 119883 (ℎ1)119883 (ℎ

2) minus ⟨ℎ

1 ℎ

2⟩H119879

(33)

6 International Journal of Stochastic Analysis

The following proposition shows that our approach todefine multiple Wiener integrals is consistent with the tra-ditional approach where multiple Wiener integrals for moregeneral Gaussian process 119883 are defined as the closed linearspace generated by Hermite polynomials

Proposition 27 Let 119867119901be the 119901th Hermite polynomial and

let ℎ isinH119879 Then

119868119879119901(ℎ

otimes119901) = 119901 ℎ

119901

H119879119867119901(119883 (ℎ)

ℎH119879

) (34)

Proof First note that without loss of generalitywe can assumeℎH119879

= 1Now by the definition of multipleWiener integralwith respect to119883 we have

119868119879119901(ℎ

otimes119901) = 119868

119882

119879119901(119870

lowast

119879119901ℎotimes119901) (35)

where

119870lowast

119879119901ℎotimes119901= (119870

lowast

119879ℎ)

otimes119901

(36)

Consequently by [22 Proposition 114] we obtain

119868119879119901(ℎ

otimes119901) = 119901119867

119901(119882 (119870

lowast

119879ℎ)) (37)

which implies the result together withTheorem 22

Proposition 27 extends to the following product formulawhich is also well-known in the Gaussianmartingale case butapparently new for general Gaussian processes Again theproof is straightforward application of transfer principle

Proposition 28 Let 119891 isinH119879119901

and 119892 isinH119879119902Then

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimes

119870119879119903119892)

(38)

where

119891otimes119870119879119903119892 = (119870

lowast

119879119901+119902minus2119903)minus1

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (39)

and (119870lowast

119879119901+119902minus2119903)minus1 denotes the preimage of 119870lowast

119879119901+119902minus2119903

Proof The proof follows directly from the definition of119868119879119901(119891) and [22 Proposition 113]

Example 29 Let 119891 isin H119879119901

and 119892 isin H119879119902

be of forms119891(119909

1 119909

119901) = prod

119901

119896=1119891119896(119909

119896) and119892(119910

1 119910

119901) = prod

119902

119896=1119892119896(119910

119896)

Then

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (119909

1 119909

119901minus119903 119910

1 119910

119902minus119903)

= int[0119879]119903

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119903

prod

119895=1

119870lowast

119879119891119901minus119903+119895

(119904119895)

sdot

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot

119903

prod

119895=1

119870lowast

119879119892119902minus119903+119895(119904

119895) d119904

1sdot sdot sdot d119904

119903

=

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot ⟨

119903

prod

119895=1

119891119901minus119903+119895

119903

prod

119895=1

119892119902minus119903+119895⟩

H119879119903

(40)

Hence119891otimes

119870119879119903119892

=

119901minus119903

prod

119896=1

119891119896(119909

119896)

119902minus119903

prod

119896=1

119892119896(119910

119896)⟨

119903

prod

119895=1

119891119895

119903

prod

119895=1

119892119895⟩

H119879119903

(41)

Remark 30 In the literature multiple Wiener integrals areusually defined as the closed linear space spanned byHermitepolynomials In such a case Proposition 27 is clearly true bythe very definition Furthermore one has a multiplicationformula (see eg [23])

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimesH119879119903

119892)

(42)

where 119891otimesH119879119903119892 denotes symmetrization of tensor product

119891otimesH119879119903119892 =

infin

sum

1198941 119894119903=1

⟨119891 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

otimes ⟨119892 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

(43)

and 119890119896 119896 = 1 is a complete orthonormal basis of the

Hilbert space H119879 Clearly by Proposition 27 both formulas

coincide This also shows that (39) is well-defined

43 Malliavin Calculus and Skorohod Integrals We begin byrecalling some basic facts on Malliavin calculus

Definition 31 Denote by S the space of all smooth randomvariables of the form

119865 = 119891 (119883 (1205931) 119883 (120593

119899)) 120593

1 120593

119899isinH

119879 (44)

International Journal of Stochastic Analysis 7

where 119891 isin 119862infin119887(R119899) that is 119891 and all its derivatives are

bounded The Malliavin derivative 119863119879119865 = 119863

119883

119879119865 of 119865 is an

element of 1198712(ΩH119879) defined by

119863119879119865 =

119899

sum

119894=1

120597119894119891 (119883 (120593

1) 119883 (120593

119899)) 120593

119894 (45)

In particular119863119879119883119905= 1

119905

Definition 32 Let D12= D12

119883be the Hilbert space of all

square integrable Malliavin differentiable random variablesdefined as the closure of S with respect to norm

1198652

12= E [|119865|

2] + E [

10038171003817100381710038171198631198791198651003817100381710038171003817

2

H119879] (46)

The divergence operator 120575119879is defined as the adjoint

operator of the Malliavin derivative119863119879

Definition 33 The domain Dom 120575119879of the operator 120575

119879is the

set of random variables 119906 isin 1198712(ΩH119879) satisfying

10038161003816100381610038161003816E ⟨119863

119879119865 119906⟩

H119879

10038161003816100381610038161003816le 119888

119906119865

1198712 (47)

for any 119865 isin D12 and some constant 119888119906depending only on 119906

For 119906 isin Dom 120575119879the divergence operator 120575

119879(119906) is a square

integrable random variable defined by the duality relation

E [119865120575119879(119906)] = E ⟨119863

119879119865 119906⟩

H119879(48)

for all 119865 isin D12

Remark 34 It is well-known that D12sub Dom 120575

119879

We use the notation

120575119879(119906) = int

119879

0

119906119904120575119883

119904 (49)

Theorem 35 (transfer principle for Malliavin calculus) Let119883 be a separable centered Gaussian process with Fredholmrepresentation (8) Let 119863

119879and 120575

119879be the Malliavin derivative

and the Skorohod integral with respect to119883 on [0 119879] Similarlylet 119863119882

119879and 120575119882

119879be the Malliavin derivative and the Skorohod

integral with respect to the Brownianmotion119882 of (8) restrictedon [0 119879] Then

120575119879= 120575

119882

119879119870lowast

119879

119870lowast

119879119863119879= 119863

119882

119879

(50)

Proof The proof follows directly from transfer principle andthe isometry provided by 119870lowast

119879with the same arguments as in

[1] Indeed by isometry we have

H119879= (119870

lowast

119879)minus1

(1198712([0 119879])) (51)

where (119870lowast

119879)minus1 denotes the preimage which implies that

D12(H

119879) = (119870

lowast

119879)minus1

(D12

119882(119871

2([0 119879]))) (52)

which justifies 119870lowast

119879119863119879= 119863

119882

119879 Furthermore we have relation

E ⟨119906119863119879119865⟩

H119879= E ⟨119870

lowast

119879119906119863

119882

119879119865⟩

1198712([0119879])

(53)

for any smooth random variable 119865 and 119906 isin 1198712(ΩH119879)

Hence by the very definition ofDom 120575 and transfer principlewe obtain

Dom 120575119879= (119870

lowast

119879)minus1

(Dom 120575119882119879) (54)

and 120575119879(119906) = 120575

119882

119879(119870

lowast

119879119906) proving the claim

Now we are ready to show that the definition of themultiple Wiener integral 119868

119879119901in Section 42 is correct in the

sense that it agrees with the iterated Skorohod integral

Proposition 36 Let ℎ isin H119879119901

be of form ℎ(1199091 119909

119901) =

prod119901

119896=1ℎ119896(119909

119896) Then ℎ is iteratively 119901 times Skorohod integrable

and

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901) 120575119883

1199051sdot sdot sdot 120575119883

119905119901= 119868

119879119901(ℎ) (55)

Moreover if ℎ isin H0

119879119901is such that it is 119901 times iteratively

Skorohod integrable then (55) still holds

Proof Again the idea is to use the transfer principle togetherwith induction Note first that the statement is true for 119901 = 1by definition and assume next that the statement is validfor 119896 = 1 119901 We denote 119891

119895= prod

119895

119896=1ℎ119896(119909

119896) Hence by

induction assumption we have

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901 119905V) 1205751198831199051

sdot sdot sdot 120575119883119905119901120575119883V

= int

119879

0

119868119879119901(119891

119901) ℎ

119901+1(V) 120575119883V

(56)

Put now 119865 = 119868119879119901(119891

119901) and 119906(119905) = ℎ

119901+1(119905) Hence by [22

Proposition 133] and by applying the transfer principle weobtain that 119865119906 belongs to Dom 120575

119879and

120575119879(119865119906) = 120575

119879(119906) 119865 minus ⟨119863

119905119865 119906 (119905)⟩

H119879

= 119868119882

119879(119870

lowast

119879ℎ119901+1) 119868

119882

119879119901(119870

lowast

119879119901119891119901)

minus 119901 ⟨119868119879119901minus1(119891

119901(sdot 119905)) ℎ

119901+1(119905)⟩

H119879

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119882

119879119901minus1(119870

lowast

119879119901minus1119891119901otimes1119870lowast

119879ℎ119901+1)

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119879119901minus1(119891

119901otimes1198701198791ℎ119901+1)

(57)

Hence the result is valid also for 119901+ 1 by Proposition 28 with119902 = 1

The claim for general ℎ isinH0

119879119901follows by approximating

with a product of simple function

8 International Journal of Stochastic Analysis

Remark 37 Note that (55) does not hold for arbitrary ℎ isinH0

119879119901in general without the a priori assumption of 119901 times

iterative Skorohod integrability For example let 119901 = 2 and119883 = 119861

119867 be a fractional Brownian motion with119867 le 14 anddefine ℎ

119905(119904 V) = 1

119905(119904)1

119904(V) for some fixed 119905 isin [0 119879] Then

int

119879

0

ℎ119905(119904 V) 120575119883V = 119883119904

1119905(119904) (58)

But119883sdot1119905does not belong to Dom 120575

119879(see [7])

We end this section by providing an extension of Ito for-mulas provided by Alos et al [1] They considered GaussianVolterra processes that is they assumed the representation

119883119905= int

119905

0

119870 (119905 119904) d119882119904 (59)

where the kernel119870 satisfied certain technical assumptions In[1] it was proved that in the case of Volterra processes one has

119891 (119883119905) = 119891 (0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(60)

if 119891 satisfies the growth condition

max [1003816100381610038161003816119891 (119909)1003816100381610038161003816 100381610038161003816100381610038161198911015840(119909)100381610038161003816100381610038161003816100381610038161003816100381611989110158401015840(119909)10038161003816100381610038161003816] le 119888119890

120582|119909|2

(61)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1 In the

following we will consider different approach which enablesus to

(i) prove that such formula holds with minimal require-ments

(ii) give more instructive proof of such result(iii) extend the result from Volterra context to more

general Gaussian processes(iv) drop some technical assumptions posed in [1]

For simplicity we assume that the variance of 119883 is ofbounded variation to guarantee the existence of the integral

int

119879

0

11989110158401015840(119883

119905) d119877 (119905 119905) (62)

If the variance is not of bounded variation then integral (62)may be understood by integration by parts if 11989110158401015840 is smoothenough or in the general case via the inner product ⟨sdot sdot⟩H119879 InTheorem 40 we also have to assume that the variance of119883is bounded

The result for polynomials is straightforward once weassume that the paths of polynomials of 119883 belong to1198712(ΩH

119879)

Proposition 38 (Ito formula for polynomials) Let 119883 be aseparable centered Gaussian process with covariance 119877 andassume that 119901 is a polynomial Furthermore assume that for

each polynomial 119901 one has 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) Then for

each 119905 isin [0 119879] one has

119901 (119883119905) = 119901 (119883

0) + int

119905

0

1199011015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11990110158401015840(119883

119904) d119877 (119904 119904)

(63)

if and only if119883sdot1119905belongs to Dom 120575

119879

Remark 39 The message of the above result is that once theprocesses 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) then they automatically

belong to the domain of 120575119879which is a subspace of 1198712(ΩH

119879)

However in order to check 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) one needs

more information on the kernel 119870119879 A sufficient condition is

provided in Corollary 43 which covers many cases of interest

Proof By definition and applying transfer principle we haveto prove that 1199011015840(119883

sdot)1

119905belongs to domain of 120575

119879and that

Eint119905

0

119863119904119866119870

lowast

119879[119901

1015840(119883

sdot) 1

119905] d119904

= E [119866119901 (119883119905)] minus E [119866119901 (119883

0)]

minus1

2int

119905

0

E [11986611990110158401015840(119883

119905)] d119877 (119905 119905)

(64)

for every random variable 119866 from a total subset of 1198712(Ω)In other words it is sufficient to show that (69) is valid forrandom variables of form 119866 = 119868119882

119899(ℎ

otimes119899) where ℎ is a step-

functionNote first that it is sufficient to prove the claim only for

Hermite polynomials119867119896 119896 = 1 Indeed it is well-known

that any polynomial can be expressed as a linear combinationof Hermite polynomials and consequently the result forarbitrary polynomial 119901 follows by linearity

We proceed by induction First it is clear that firsttwo polynomials 119867

0and 119867

1satisfy (64) Furthermore by

assumption 1198671015840

2(119883

sdot)1

119905belongs to Dom 120575

119879from which (64)

is easily deduced by [22 Proposition 133] Assume nextthat the result is valid for Hermite polynomials 119867

119896 119896 =

0 1 119899 Then recall well-known recursion formulas

119867119899+1(119909) = 119909119867

119899(119909) minus 119899119867

119899minus1(119909)

1198671015840

119899(119909) = 119899119867

119899minus1(119909)

(65)

The induction step follows with straightforward calcula-tions by using the recursion formulas above and [22Proposition 133] We leave the details to the reader

We will now illustrate how the result can be generalizedfor functions satisfying the growth condition (61) by usingProposition 38 First note that the growth condition (61) isindeed natural since it guarantees that the left side of (60)is square integrable Consequently since operator 120575

119879is a

mapping from 1198712(ΩH119879) into 1198712(Ω) functions satisfying

(61) are the largest class of functions for which (60) can holdHowever it is not clear in general whether 1198911015840(119883

sdot)1

119905belongs

International Journal of Stochastic Analysis 9

to Dom 120575119879 Indeed for example in [1] the authors posed

additional conditions on the Volterra kernel 119870 to guaranteethis As our main result we show that E1198911015840(119883

sdot)1

1199052

H119879lt infin

implies that (60) holds In other words not only is the Itoformula (60) natural but it is also the only possibility

Theorem 40 (Ito formula for Skorohod integrals) Let 119883 bea separable centered Gaussian process with covariance 119877 suchthat all the polynomials 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) Assume that

119891 isin 1198622 satisfies growth condition (61) and that the variance of

119883 is bounded and of bounded variation If

E100381710038171003817100381710038171198911015840(119883

sdot) 1

119905

10038171003817100381710038171003817

2

H119879lt infin (66)

for any 119905 isin [0 119879] then

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(67)

Proof In this proof we assume for notational simplicity andwith no loss of generality that sup

0le119904le119879119877(119904 119904) = 1

First it is clear that (66) implies that 1198911015840(119883sdot)1

119905belongs to

domain of 120575119879 Hence we only have to prove that

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E [119866119891 (119883119905)] minus E [119866119891 (119883

0)]

minus1

2int

119905

0

E [11986611989110158401015840(119883

119904)] d119877 (119904 119904)

(68)

for every random variable 119866 = 119868119882119899(ℎ

otimes119899)

Now it is well-known that Hermite polynomials whenproperly scaled form an orthogonal system in 1198712(R) whenequipped with the Gaussian measure Now each 119891 satisfyingthe growth condition (61) has a series representation

119891 (119909) =

infin

sum

119896=0

120572119896119867119896(119909) (69)

Indeed the growth condition (61) implies that

intR

100381610038161003816100381610038161198911015840(119909)10038161003816100381610038161003816

2

119890minus1199092(2sup

0le119904le119879119877(119904119904))d119909 lt infin (70)

Furthermore we have

119891 (119883119904) =

infin

sum

119896=0

120572119896119867119896(119883

119904) (71)

where the series converge almost surely and in 1198712(Ω) andsimilar conclusion is valid for derivatives1198911015840(119883

119904) and 11989110158401015840(119883

119904)

Then by applying (66) we obtain that for any 120598 gt 0 thereexists119873 = 119873

120598such that we have

E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598 119899 ge 119873 (72)

where

1198911015840

119899(119883

119904) =

infin

sum

119896=119899

120572119896119867

1015840

119896(119883

119904) (73)

Consequently for random variables of form 119866 = 119868119882119899(ℎ

otimes119899) we

obtain by choosing 119873 large enough and applying Proposi-tion 38 that

E [119866119891 (119883119905)] minus E [119866119891 (119883

0)] minus1

2

sdot int

119905

0

E (11986611989110158401015840(119883

119905)) d119877 (119905 119905)

minus E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598

(74)

Now the left side does not depend on 119899 which concludes theproof

Remark 41 Note that actually it is sufficient to have

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

=

infin

sum

119896=1

120572119896E ⟨119863

119879119866119867

1015840

119896(119883

sdot) 1

119905⟩H119879

(75)

from which the result follows by Proposition 38 Further-more taking account growth condition (61) this is actuallysufficient and necessary condition for formula (60) to holdConsequently our method can also be used to obtain Itoformulas by considering extended domain of 120575

119879(see [9] or

[11]) This is the topic of Section 44

Example 42 It is known that if 119883 = 119861119867 is a fractional Brow-nian motion with119867 gt 14 then 1198911015840(119883

sdot)1

119905satisfies condition

(66) while for 119867 le 14 it does not (see [22 Chapter 5])Consequently a simple application of Theorem 40 coversfractional Brownian motion with 119867 gt 14 For the case119867 le 14 one has to consider extended domain of 120575

119879which is

proved in [9] Consequently in this case we have (75) for any119865 isin S

We end this section by illustrating the power of ourmethod with the following simple corollary which is anextension of [1 Theorem 1]

Corollary 43 Let 119883 be a separable centered continuousGaussian process with covariance 119877 that is bounded such thatthe Fredholm kernel 119870

119879is of bounded variation and

int

119879

0

(int

119879

0

1003817100381710038171003817119883119905minus 119883

119904

10038171003817100381710038171198712(Ω)

10038161003816100381610038161198701198791003816100381610038161003816 (d119905 119904))

2

d119904 lt infin (76)

Then for any 119905 isin [0 119879] one has

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(77)

10 International Journal of Stochastic Analysis

Proof Note that assumption is a Fredholm version of con-dition (K2) in [1] which implies condition (66) Hence theresult follows byTheorem 40

44 Extended Divergence Operator As shown in Section 43the Ito formula (60) is the only possibility However theproblem is that the space 1198712(ΩH

119879) may be too small to

contain the elements 1198911015840(119883sdot)1

119905 In particular it may happen

that not even the process 119883 itself belongs to 1198712(ΩH119879) (see

eg [7] for the case of fractional Brownian motion with119867 le 14) This problem can be overcome by consideringan extended domain of 120575

119879 The idea of extended domain is

to extend the inner product ⟨119906 120593⟩H119879 for simple 120593 to moregeneral processes 119906 and then define extended domain by(47) with a restricted class of test variables 119865 This also givesanother intuitive reason why extended domain of 120575

119879can be

useful indeed here we have proved that Ito formula (60) isthe only possibility and what one essentially needs for suchresult is the following

(i) 119883sdot1119905belongs to Dom 120575

119879

(ii) Equation (75) is valid for functions satisfying (61)

Consequently one should look for extensions of operator 120575119879

such that these two things are satisfiedTo facilitate the extension of domain we make the

following relatively moderate assumption

(H) The function 119905 997891rarr 119877(119905 119904) is of bounded variation on[0 119879] and

sup119905isin[0119879]

int

119879

0

|119877| (d119904 119905) lt infin (78)

Remark 44 Note that we are making the assumption on thecovariance 119877 not the kernel 119870

119879 Hence our case is different

from that of [1] Also [11] assumed absolute continuity in 119877we are satisfied with bounded variation

We will follow the idea from Lei and Nualart [11] andextend the inner product ⟨sdot sdot⟩H119879 beyondH

119879

Consider a step-function 120593Then on the one hand by theisometry property we have

⟨120593 1119905⟩H119879= int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 (79)

where 119892119905(119904) = 119870(119905 119904) isin 119871

2([0 119879]) On the other hand by

using adjoint property (see Remark 20) we obtain

int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 = int

119879

0

120593 (119904) (119870119879119892119905) (d119904) (80)

where computing formally we have

(119870119879119892119905) (d119904) = int

119879

0

119892119905(119906)119870

119879(d119904 119906) d119906

= int

119879

0

119870119879(119905 119906)119870

119879(d119904 119906) d119906 = 119877 (119905 d119904)

(81)

Consequently

⟨120593 1119905⟩H119879= int

119879

0

120593 (119904) 119877 (119905 d119904) (82)

This gives motivation to the following definition similar tothat of [11 Definition 21]

Definition 45 Denote by T119879the space of measurable func-

tions 119892 satisfying

sup119905isin[0119879]

int

119879

0

1003816100381610038161003816119892 (119904)1003816100381610038161003816 |119877| (119905 d119904) lt infin (83)

and let 120593 be a step-function of form 120593 = sum119899119896=11198871198961119905119896 Then we

extend ⟨sdot sdot⟩H119879 toT119879by defining

⟨119892 120593⟩H119879=

119899

sum

119896=1

119887119896int

119879

0

119892 (119904) 119877 (119905119896 d119904) (84)

In particular this implies that for 119892 and 120593 as above we have

⟨1198921119905 120593⟩

H119879= int

119905

0

119892 (119904) d ⟨1119904 120593⟩

H119879 (85)

We define extended domain Dom119864120575119879similarly as in [11]

Definition 46 A process 119906 isin 1198711(ΩT119879) belongs to Dom119864

120575119879

if10038161003816100381610038161003816E ⟨119906119863119865⟩H119879

10038161003816100381610038161003816le 119888

119906119865

2(86)

for any smooth random variable 119865 isin S In this case 120575(119906) isin1198712(Ω) is defined by duality relationship

E [119865120575 (119906)] = E ⟨119906119863119865⟩H119879 (87)

Remark 47 Note that in general Dom 120575119879and Dom119864

120575119879are

not comparable See [11] for discussion

Note now that if a function 119891 satisfies the growthcondition (61) then 1198911015840(119883

sdot)1

119905isin 119871

1(ΩT

119879) since (61) implies

E sup119905isin[0119879]

100381610038161003816100381610038161198911015840(119883

119905)10038161003816100381610038161003816

119901

lt infin (88)

for any 119901 lt (12120582)(sup119905isin[0119879]

119877(119905 119905))minus1 Consequently with

this definition we are able to get rid of the problem thatprocesses might not belong to corresponding H

119879-spaces

Furthermore this implies that the series expansion (69)converges in the norm 1198711(ΩT

119879) defined by

Eint119879

0

|119906 (119904)| |119877| (119905 d119904) (89)

which in turn implies (75) Hence it is straightforward toobtain the following by first showing the result for polynomi-als and then by approximating in a similar manner as done inSection 43 but using the extended domain instead

International Journal of Stochastic Analysis 11

Theorem 48 (Ito formula for extended Skorohod integrals)Let119883 be a separable centered Gaussian process with covariance119877 and assume that 119891 isin 1198622 satisfies growth condition (61)Furthermore assume that (H) holds and that the variance of119883is bounded and is of bounded variationThen for any 119905 isin [0 119879]the process 1198911015840(119883

sdot)1

119905belongs to Dom119864

120575119879and

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(90)

Remark 49 As an application of Theorem 48 it is straight-forward to derive version of Ito-Tanaka formula underadditional conditions which guarantee that for a certainsequence of functions 119891

119899we have the convergence of term

(12) int119905

011989110158401015840

119899(119883

119904)d119877(119904 119904) to the local time For details we refer

to [11] where authors derived such formula under theirassumptions

Finally let us note that the extension to functions 119891(119905 119909)is straightforward where 119891 satisfies the following growthcondition

max [1003816100381610038161003816119891 (119905 119909)1003816100381610038161003816 1003816100381610038161003816120597119905119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119909119891 (119905 119909)

1003816100381610038161003816]

le 119888119890120582|119909|2

(91)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1

Theorem 50 (Ito formula for extended Skorohod integralsII) Let 119883 be a separable centered Gaussian process withcovariance 119877 and assume that 119891 isin 11986212 satisfies growthcondition (91) Furthermore assume that (H) holds and thatthe variance of119883 is bounded and is of bounded variationThenfor any 119905 isin [0 119879] the process 120597

119909119891(sdot 119883

sdot)1

119905belongs to Dom119864

120575119879

and

119891 (119905 119883119905) = 119891 (0 119883

0) + int

119905

0

120597119909119891 (119904 119883

119904) 120575119883

119904

+ int

119905

0

120597119905119891 (119904 119883

119904) d119904

+1

2int

119905

0

120597119909119909119891 (119904 119883

119904) d119877 (119904 119904)

(92)

Proof Taking into account that we have no problems con-cerning processes to belong to the required spaces theformula follows by approximating with polynomials of form119901(119909)119902(119905) and following the proof of Theorem 40

5 Applications

We illustrate how some results transfer easily from theBrownian case to the Gaussian Fredholm processes

51 Equivalence in Law The transfer principle has alreadybeen used in connection with the equivalence of law of

Gaussian processes in for example [24] in the context offractional Brownian motions and in [2] in the context ofGaussianVolterra processes satisfying certain nondegeneracyconditions The following proposition uses the Fredholmrepresentation (8) to give a sufficient condition for theequivalence of general Gaussian processes in terms of theirFredholm kernels

Proposition 51 Let 119883 and be two Gaussian processes withFredholm kernels 119870

119879and

119879 respectively If there exists a

Volterra kernel ℓ isin 1198712([0 119879]2) such that

119879(119905 119904) = 119870

119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 (93)

then119883 and are equivalent in law

Proof Recall that by the Hitsuda representation theorem [5Theorem 63] a centered Gaussian process is equivalent inlaw to a Brownian motion on [0 119879] if and only if there existsa kernel ℓ isin 1198712([0 119879]2) and a Brownian motion119882 such that admits the representation

119905= 119882

119905minus int

119905

0

int

119904

0

ℓ (119904 119906) d119882119906d119904 (94)

Let119883 have the Fredholm representations

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (95)

Then is equivalent to119883 if it admits in law the representa-tion

119905

119889

= int

119879

0

119870119879(119905 119904) d

119904 (96)

where is connected to119882 of (95) by (94)In order to show (96) let

119905= int

119879

0

119879(119905 119904) d1198821015840

119905(97)

be the Fredholm representation of Here 1198821015840 is someBrownian motion Then by using connection (93) and theFubini theorem we obtain

119905= int

119879

0

119879(119905 119904) d1198821015840

119904

119889

= int

119879

0

119879(119905 119904) d119882

119904

= int

119879

0

(119870119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906) d119882

119904

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

6 International Journal of Stochastic Analysis

The following proposition shows that our approach todefine multiple Wiener integrals is consistent with the tra-ditional approach where multiple Wiener integrals for moregeneral Gaussian process 119883 are defined as the closed linearspace generated by Hermite polynomials

Proposition 27 Let 119867119901be the 119901th Hermite polynomial and

let ℎ isinH119879 Then

119868119879119901(ℎ

otimes119901) = 119901 ℎ

119901

H119879119867119901(119883 (ℎ)

ℎH119879

) (34)

Proof First note that without loss of generalitywe can assumeℎH119879

= 1Now by the definition of multipleWiener integralwith respect to119883 we have

119868119879119901(ℎ

otimes119901) = 119868

119882

119879119901(119870

lowast

119879119901ℎotimes119901) (35)

where

119870lowast

119879119901ℎotimes119901= (119870

lowast

119879ℎ)

otimes119901

(36)

Consequently by [22 Proposition 114] we obtain

119868119879119901(ℎ

otimes119901) = 119901119867

119901(119882 (119870

lowast

119879ℎ)) (37)

which implies the result together withTheorem 22

Proposition 27 extends to the following product formulawhich is also well-known in the Gaussianmartingale case butapparently new for general Gaussian processes Again theproof is straightforward application of transfer principle

Proposition 28 Let 119891 isinH119879119901

and 119892 isinH119879119902Then

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimes

119870119879119903119892)

(38)

where

119891otimes119870119879119903119892 = (119870

lowast

119879119901+119902minus2119903)minus1

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (39)

and (119870lowast

119879119901+119902minus2119903)minus1 denotes the preimage of 119870lowast

119879119901+119902minus2119903

Proof The proof follows directly from the definition of119868119879119901(119891) and [22 Proposition 113]

Example 29 Let 119891 isin H119879119901

and 119892 isin H119879119902

be of forms119891(119909

1 119909

119901) = prod

119901

119896=1119891119896(119909

119896) and119892(119910

1 119910

119901) = prod

119902

119896=1119892119896(119910

119896)

Then

(119870lowast

119879119901119891otimes

119903119870lowast

119879119902119892) (119909

1 119909

119901minus119903 119910

1 119910

119902minus119903)

= int[0119879]119903

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119903

prod

119895=1

119870lowast

119879119891119901minus119903+119895

(119904119895)

sdot

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot

119903

prod

119895=1

119870lowast

119879119892119902minus119903+119895(119904

119895) d119904

1sdot sdot sdot d119904

119903

=

119901minus119903

prod

119896=1

119870lowast

119879119891119896(119909

119896)

119902minus119903

prod

119896=1

119870lowast

119879119892119896(119910

119896)

sdot ⟨

119903

prod

119895=1

119891119901minus119903+119895

119903

prod

119895=1

119892119902minus119903+119895⟩

H119879119903

(40)

Hence119891otimes

119870119879119903119892

=

119901minus119903

prod

119896=1

119891119896(119909

119896)

119902minus119903

prod

119896=1

119892119896(119910

119896)⟨

119903

prod

119895=1

119891119895

119903

prod

119895=1

119892119895⟩

H119879119903

(41)

Remark 30 In the literature multiple Wiener integrals areusually defined as the closed linear space spanned byHermitepolynomials In such a case Proposition 27 is clearly true bythe very definition Furthermore one has a multiplicationformula (see eg [23])

119868119879119901(119891) 119868

119879119902(119892)

=

119901and119902

sum

119903=0

119903 (

119901

119903)(

119902

119903) 119868

119879119901+119902minus2119903(119891otimesH119879119903

119892)

(42)

where 119891otimesH119879119903119892 denotes symmetrization of tensor product

119891otimesH119879119903119892 =

infin

sum

1198941 119894119903=1

⟨119891 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

otimes ⟨119892 1198901198941otimes sdot sdot sdot otimes 119890

119894119903⟩Hotimes119903119879

(43)

and 119890119896 119896 = 1 is a complete orthonormal basis of the

Hilbert space H119879 Clearly by Proposition 27 both formulas

coincide This also shows that (39) is well-defined

43 Malliavin Calculus and Skorohod Integrals We begin byrecalling some basic facts on Malliavin calculus

Definition 31 Denote by S the space of all smooth randomvariables of the form

119865 = 119891 (119883 (1205931) 119883 (120593

119899)) 120593

1 120593

119899isinH

119879 (44)

International Journal of Stochastic Analysis 7

where 119891 isin 119862infin119887(R119899) that is 119891 and all its derivatives are

bounded The Malliavin derivative 119863119879119865 = 119863

119883

119879119865 of 119865 is an

element of 1198712(ΩH119879) defined by

119863119879119865 =

119899

sum

119894=1

120597119894119891 (119883 (120593

1) 119883 (120593

119899)) 120593

119894 (45)

In particular119863119879119883119905= 1

119905

Definition 32 Let D12= D12

119883be the Hilbert space of all

square integrable Malliavin differentiable random variablesdefined as the closure of S with respect to norm

1198652

12= E [|119865|

2] + E [

10038171003817100381710038171198631198791198651003817100381710038171003817

2

H119879] (46)

The divergence operator 120575119879is defined as the adjoint

operator of the Malliavin derivative119863119879

Definition 33 The domain Dom 120575119879of the operator 120575

119879is the

set of random variables 119906 isin 1198712(ΩH119879) satisfying

10038161003816100381610038161003816E ⟨119863

119879119865 119906⟩

H119879

10038161003816100381610038161003816le 119888

119906119865

1198712 (47)

for any 119865 isin D12 and some constant 119888119906depending only on 119906

For 119906 isin Dom 120575119879the divergence operator 120575

119879(119906) is a square

integrable random variable defined by the duality relation

E [119865120575119879(119906)] = E ⟨119863

119879119865 119906⟩

H119879(48)

for all 119865 isin D12

Remark 34 It is well-known that D12sub Dom 120575

119879

We use the notation

120575119879(119906) = int

119879

0

119906119904120575119883

119904 (49)

Theorem 35 (transfer principle for Malliavin calculus) Let119883 be a separable centered Gaussian process with Fredholmrepresentation (8) Let 119863

119879and 120575

119879be the Malliavin derivative

and the Skorohod integral with respect to119883 on [0 119879] Similarlylet 119863119882

119879and 120575119882

119879be the Malliavin derivative and the Skorohod

integral with respect to the Brownianmotion119882 of (8) restrictedon [0 119879] Then

120575119879= 120575

119882

119879119870lowast

119879

119870lowast

119879119863119879= 119863

119882

119879

(50)

Proof The proof follows directly from transfer principle andthe isometry provided by 119870lowast

119879with the same arguments as in

[1] Indeed by isometry we have

H119879= (119870

lowast

119879)minus1

(1198712([0 119879])) (51)

where (119870lowast

119879)minus1 denotes the preimage which implies that

D12(H

119879) = (119870

lowast

119879)minus1

(D12

119882(119871

2([0 119879]))) (52)

which justifies 119870lowast

119879119863119879= 119863

119882

119879 Furthermore we have relation

E ⟨119906119863119879119865⟩

H119879= E ⟨119870

lowast

119879119906119863

119882

119879119865⟩

1198712([0119879])

(53)

for any smooth random variable 119865 and 119906 isin 1198712(ΩH119879)

Hence by the very definition ofDom 120575 and transfer principlewe obtain

Dom 120575119879= (119870

lowast

119879)minus1

(Dom 120575119882119879) (54)

and 120575119879(119906) = 120575

119882

119879(119870

lowast

119879119906) proving the claim

Now we are ready to show that the definition of themultiple Wiener integral 119868

119879119901in Section 42 is correct in the

sense that it agrees with the iterated Skorohod integral

Proposition 36 Let ℎ isin H119879119901

be of form ℎ(1199091 119909

119901) =

prod119901

119896=1ℎ119896(119909

119896) Then ℎ is iteratively 119901 times Skorohod integrable

and

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901) 120575119883

1199051sdot sdot sdot 120575119883

119905119901= 119868

119879119901(ℎ) (55)

Moreover if ℎ isin H0

119879119901is such that it is 119901 times iteratively

Skorohod integrable then (55) still holds

Proof Again the idea is to use the transfer principle togetherwith induction Note first that the statement is true for 119901 = 1by definition and assume next that the statement is validfor 119896 = 1 119901 We denote 119891

119895= prod

119895

119896=1ℎ119896(119909

119896) Hence by

induction assumption we have

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901 119905V) 1205751198831199051

sdot sdot sdot 120575119883119905119901120575119883V

= int

119879

0

119868119879119901(119891

119901) ℎ

119901+1(V) 120575119883V

(56)

Put now 119865 = 119868119879119901(119891

119901) and 119906(119905) = ℎ

119901+1(119905) Hence by [22

Proposition 133] and by applying the transfer principle weobtain that 119865119906 belongs to Dom 120575

119879and

120575119879(119865119906) = 120575

119879(119906) 119865 minus ⟨119863

119905119865 119906 (119905)⟩

H119879

= 119868119882

119879(119870

lowast

119879ℎ119901+1) 119868

119882

119879119901(119870

lowast

119879119901119891119901)

minus 119901 ⟨119868119879119901minus1(119891

119901(sdot 119905)) ℎ

119901+1(119905)⟩

H119879

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119882

119879119901minus1(119870

lowast

119879119901minus1119891119901otimes1119870lowast

119879ℎ119901+1)

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119879119901minus1(119891

119901otimes1198701198791ℎ119901+1)

(57)

Hence the result is valid also for 119901+ 1 by Proposition 28 with119902 = 1

The claim for general ℎ isinH0

119879119901follows by approximating

with a product of simple function

8 International Journal of Stochastic Analysis

Remark 37 Note that (55) does not hold for arbitrary ℎ isinH0

119879119901in general without the a priori assumption of 119901 times

iterative Skorohod integrability For example let 119901 = 2 and119883 = 119861

119867 be a fractional Brownian motion with119867 le 14 anddefine ℎ

119905(119904 V) = 1

119905(119904)1

119904(V) for some fixed 119905 isin [0 119879] Then

int

119879

0

ℎ119905(119904 V) 120575119883V = 119883119904

1119905(119904) (58)

But119883sdot1119905does not belong to Dom 120575

119879(see [7])

We end this section by providing an extension of Ito for-mulas provided by Alos et al [1] They considered GaussianVolterra processes that is they assumed the representation

119883119905= int

119905

0

119870 (119905 119904) d119882119904 (59)

where the kernel119870 satisfied certain technical assumptions In[1] it was proved that in the case of Volterra processes one has

119891 (119883119905) = 119891 (0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(60)

if 119891 satisfies the growth condition

max [1003816100381610038161003816119891 (119909)1003816100381610038161003816 100381610038161003816100381610038161198911015840(119909)100381610038161003816100381610038161003816100381610038161003816100381611989110158401015840(119909)10038161003816100381610038161003816] le 119888119890

120582|119909|2

(61)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1 In the

following we will consider different approach which enablesus to

(i) prove that such formula holds with minimal require-ments

(ii) give more instructive proof of such result(iii) extend the result from Volterra context to more

general Gaussian processes(iv) drop some technical assumptions posed in [1]

For simplicity we assume that the variance of 119883 is ofbounded variation to guarantee the existence of the integral

int

119879

0

11989110158401015840(119883

119905) d119877 (119905 119905) (62)

If the variance is not of bounded variation then integral (62)may be understood by integration by parts if 11989110158401015840 is smoothenough or in the general case via the inner product ⟨sdot sdot⟩H119879 InTheorem 40 we also have to assume that the variance of119883is bounded

The result for polynomials is straightforward once weassume that the paths of polynomials of 119883 belong to1198712(ΩH

119879)

Proposition 38 (Ito formula for polynomials) Let 119883 be aseparable centered Gaussian process with covariance 119877 andassume that 119901 is a polynomial Furthermore assume that for

each polynomial 119901 one has 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) Then for

each 119905 isin [0 119879] one has

119901 (119883119905) = 119901 (119883

0) + int

119905

0

1199011015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11990110158401015840(119883

119904) d119877 (119904 119904)

(63)

if and only if119883sdot1119905belongs to Dom 120575

119879

Remark 39 The message of the above result is that once theprocesses 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) then they automatically

belong to the domain of 120575119879which is a subspace of 1198712(ΩH

119879)

However in order to check 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) one needs

more information on the kernel 119870119879 A sufficient condition is

provided in Corollary 43 which covers many cases of interest

Proof By definition and applying transfer principle we haveto prove that 1199011015840(119883

sdot)1

119905belongs to domain of 120575

119879and that

Eint119905

0

119863119904119866119870

lowast

119879[119901

1015840(119883

sdot) 1

119905] d119904

= E [119866119901 (119883119905)] minus E [119866119901 (119883

0)]

minus1

2int

119905

0

E [11986611990110158401015840(119883

119905)] d119877 (119905 119905)

(64)

for every random variable 119866 from a total subset of 1198712(Ω)In other words it is sufficient to show that (69) is valid forrandom variables of form 119866 = 119868119882

119899(ℎ

otimes119899) where ℎ is a step-

functionNote first that it is sufficient to prove the claim only for

Hermite polynomials119867119896 119896 = 1 Indeed it is well-known

that any polynomial can be expressed as a linear combinationof Hermite polynomials and consequently the result forarbitrary polynomial 119901 follows by linearity

We proceed by induction First it is clear that firsttwo polynomials 119867

0and 119867

1satisfy (64) Furthermore by

assumption 1198671015840

2(119883

sdot)1

119905belongs to Dom 120575

119879from which (64)

is easily deduced by [22 Proposition 133] Assume nextthat the result is valid for Hermite polynomials 119867

119896 119896 =

0 1 119899 Then recall well-known recursion formulas

119867119899+1(119909) = 119909119867

119899(119909) minus 119899119867

119899minus1(119909)

1198671015840

119899(119909) = 119899119867

119899minus1(119909)

(65)

The induction step follows with straightforward calcula-tions by using the recursion formulas above and [22Proposition 133] We leave the details to the reader

We will now illustrate how the result can be generalizedfor functions satisfying the growth condition (61) by usingProposition 38 First note that the growth condition (61) isindeed natural since it guarantees that the left side of (60)is square integrable Consequently since operator 120575

119879is a

mapping from 1198712(ΩH119879) into 1198712(Ω) functions satisfying

(61) are the largest class of functions for which (60) can holdHowever it is not clear in general whether 1198911015840(119883

sdot)1

119905belongs

International Journal of Stochastic Analysis 9

to Dom 120575119879 Indeed for example in [1] the authors posed

additional conditions on the Volterra kernel 119870 to guaranteethis As our main result we show that E1198911015840(119883

sdot)1

1199052

H119879lt infin

implies that (60) holds In other words not only is the Itoformula (60) natural but it is also the only possibility

Theorem 40 (Ito formula for Skorohod integrals) Let 119883 bea separable centered Gaussian process with covariance 119877 suchthat all the polynomials 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) Assume that

119891 isin 1198622 satisfies growth condition (61) and that the variance of

119883 is bounded and of bounded variation If

E100381710038171003817100381710038171198911015840(119883

sdot) 1

119905

10038171003817100381710038171003817

2

H119879lt infin (66)

for any 119905 isin [0 119879] then

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(67)

Proof In this proof we assume for notational simplicity andwith no loss of generality that sup

0le119904le119879119877(119904 119904) = 1

First it is clear that (66) implies that 1198911015840(119883sdot)1

119905belongs to

domain of 120575119879 Hence we only have to prove that

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E [119866119891 (119883119905)] minus E [119866119891 (119883

0)]

minus1

2int

119905

0

E [11986611989110158401015840(119883

119904)] d119877 (119904 119904)

(68)

for every random variable 119866 = 119868119882119899(ℎ

otimes119899)

Now it is well-known that Hermite polynomials whenproperly scaled form an orthogonal system in 1198712(R) whenequipped with the Gaussian measure Now each 119891 satisfyingthe growth condition (61) has a series representation

119891 (119909) =

infin

sum

119896=0

120572119896119867119896(119909) (69)

Indeed the growth condition (61) implies that

intR

100381610038161003816100381610038161198911015840(119909)10038161003816100381610038161003816

2

119890minus1199092(2sup

0le119904le119879119877(119904119904))d119909 lt infin (70)

Furthermore we have

119891 (119883119904) =

infin

sum

119896=0

120572119896119867119896(119883

119904) (71)

where the series converge almost surely and in 1198712(Ω) andsimilar conclusion is valid for derivatives1198911015840(119883

119904) and 11989110158401015840(119883

119904)

Then by applying (66) we obtain that for any 120598 gt 0 thereexists119873 = 119873

120598such that we have

E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598 119899 ge 119873 (72)

where

1198911015840

119899(119883

119904) =

infin

sum

119896=119899

120572119896119867

1015840

119896(119883

119904) (73)

Consequently for random variables of form 119866 = 119868119882119899(ℎ

otimes119899) we

obtain by choosing 119873 large enough and applying Proposi-tion 38 that

E [119866119891 (119883119905)] minus E [119866119891 (119883

0)] minus1

2

sdot int

119905

0

E (11986611989110158401015840(119883

119905)) d119877 (119905 119905)

minus E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598

(74)

Now the left side does not depend on 119899 which concludes theproof

Remark 41 Note that actually it is sufficient to have

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

=

infin

sum

119896=1

120572119896E ⟨119863

119879119866119867

1015840

119896(119883

sdot) 1

119905⟩H119879

(75)

from which the result follows by Proposition 38 Further-more taking account growth condition (61) this is actuallysufficient and necessary condition for formula (60) to holdConsequently our method can also be used to obtain Itoformulas by considering extended domain of 120575

119879(see [9] or

[11]) This is the topic of Section 44

Example 42 It is known that if 119883 = 119861119867 is a fractional Brow-nian motion with119867 gt 14 then 1198911015840(119883

sdot)1

119905satisfies condition

(66) while for 119867 le 14 it does not (see [22 Chapter 5])Consequently a simple application of Theorem 40 coversfractional Brownian motion with 119867 gt 14 For the case119867 le 14 one has to consider extended domain of 120575

119879which is

proved in [9] Consequently in this case we have (75) for any119865 isin S

We end this section by illustrating the power of ourmethod with the following simple corollary which is anextension of [1 Theorem 1]

Corollary 43 Let 119883 be a separable centered continuousGaussian process with covariance 119877 that is bounded such thatthe Fredholm kernel 119870

119879is of bounded variation and

int

119879

0

(int

119879

0

1003817100381710038171003817119883119905minus 119883

119904

10038171003817100381710038171198712(Ω)

10038161003816100381610038161198701198791003816100381610038161003816 (d119905 119904))

2

d119904 lt infin (76)

Then for any 119905 isin [0 119879] one has

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(77)

10 International Journal of Stochastic Analysis

Proof Note that assumption is a Fredholm version of con-dition (K2) in [1] which implies condition (66) Hence theresult follows byTheorem 40

44 Extended Divergence Operator As shown in Section 43the Ito formula (60) is the only possibility However theproblem is that the space 1198712(ΩH

119879) may be too small to

contain the elements 1198911015840(119883sdot)1

119905 In particular it may happen

that not even the process 119883 itself belongs to 1198712(ΩH119879) (see

eg [7] for the case of fractional Brownian motion with119867 le 14) This problem can be overcome by consideringan extended domain of 120575

119879 The idea of extended domain is

to extend the inner product ⟨119906 120593⟩H119879 for simple 120593 to moregeneral processes 119906 and then define extended domain by(47) with a restricted class of test variables 119865 This also givesanother intuitive reason why extended domain of 120575

119879can be

useful indeed here we have proved that Ito formula (60) isthe only possibility and what one essentially needs for suchresult is the following

(i) 119883sdot1119905belongs to Dom 120575

119879

(ii) Equation (75) is valid for functions satisfying (61)

Consequently one should look for extensions of operator 120575119879

such that these two things are satisfiedTo facilitate the extension of domain we make the

following relatively moderate assumption

(H) The function 119905 997891rarr 119877(119905 119904) is of bounded variation on[0 119879] and

sup119905isin[0119879]

int

119879

0

|119877| (d119904 119905) lt infin (78)

Remark 44 Note that we are making the assumption on thecovariance 119877 not the kernel 119870

119879 Hence our case is different

from that of [1] Also [11] assumed absolute continuity in 119877we are satisfied with bounded variation

We will follow the idea from Lei and Nualart [11] andextend the inner product ⟨sdot sdot⟩H119879 beyondH

119879

Consider a step-function 120593Then on the one hand by theisometry property we have

⟨120593 1119905⟩H119879= int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 (79)

where 119892119905(119904) = 119870(119905 119904) isin 119871

2([0 119879]) On the other hand by

using adjoint property (see Remark 20) we obtain

int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 = int

119879

0

120593 (119904) (119870119879119892119905) (d119904) (80)

where computing formally we have

(119870119879119892119905) (d119904) = int

119879

0

119892119905(119906)119870

119879(d119904 119906) d119906

= int

119879

0

119870119879(119905 119906)119870

119879(d119904 119906) d119906 = 119877 (119905 d119904)

(81)

Consequently

⟨120593 1119905⟩H119879= int

119879

0

120593 (119904) 119877 (119905 d119904) (82)

This gives motivation to the following definition similar tothat of [11 Definition 21]

Definition 45 Denote by T119879the space of measurable func-

tions 119892 satisfying

sup119905isin[0119879]

int

119879

0

1003816100381610038161003816119892 (119904)1003816100381610038161003816 |119877| (119905 d119904) lt infin (83)

and let 120593 be a step-function of form 120593 = sum119899119896=11198871198961119905119896 Then we

extend ⟨sdot sdot⟩H119879 toT119879by defining

⟨119892 120593⟩H119879=

119899

sum

119896=1

119887119896int

119879

0

119892 (119904) 119877 (119905119896 d119904) (84)

In particular this implies that for 119892 and 120593 as above we have

⟨1198921119905 120593⟩

H119879= int

119905

0

119892 (119904) d ⟨1119904 120593⟩

H119879 (85)

We define extended domain Dom119864120575119879similarly as in [11]

Definition 46 A process 119906 isin 1198711(ΩT119879) belongs to Dom119864

120575119879

if10038161003816100381610038161003816E ⟨119906119863119865⟩H119879

10038161003816100381610038161003816le 119888

119906119865

2(86)

for any smooth random variable 119865 isin S In this case 120575(119906) isin1198712(Ω) is defined by duality relationship

E [119865120575 (119906)] = E ⟨119906119863119865⟩H119879 (87)

Remark 47 Note that in general Dom 120575119879and Dom119864

120575119879are

not comparable See [11] for discussion

Note now that if a function 119891 satisfies the growthcondition (61) then 1198911015840(119883

sdot)1

119905isin 119871

1(ΩT

119879) since (61) implies

E sup119905isin[0119879]

100381610038161003816100381610038161198911015840(119883

119905)10038161003816100381610038161003816

119901

lt infin (88)

for any 119901 lt (12120582)(sup119905isin[0119879]

119877(119905 119905))minus1 Consequently with

this definition we are able to get rid of the problem thatprocesses might not belong to corresponding H

119879-spaces

Furthermore this implies that the series expansion (69)converges in the norm 1198711(ΩT

119879) defined by

Eint119879

0

|119906 (119904)| |119877| (119905 d119904) (89)

which in turn implies (75) Hence it is straightforward toobtain the following by first showing the result for polynomi-als and then by approximating in a similar manner as done inSection 43 but using the extended domain instead

International Journal of Stochastic Analysis 11

Theorem 48 (Ito formula for extended Skorohod integrals)Let119883 be a separable centered Gaussian process with covariance119877 and assume that 119891 isin 1198622 satisfies growth condition (61)Furthermore assume that (H) holds and that the variance of119883is bounded and is of bounded variationThen for any 119905 isin [0 119879]the process 1198911015840(119883

sdot)1

119905belongs to Dom119864

120575119879and

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(90)

Remark 49 As an application of Theorem 48 it is straight-forward to derive version of Ito-Tanaka formula underadditional conditions which guarantee that for a certainsequence of functions 119891

119899we have the convergence of term

(12) int119905

011989110158401015840

119899(119883

119904)d119877(119904 119904) to the local time For details we refer

to [11] where authors derived such formula under theirassumptions

Finally let us note that the extension to functions 119891(119905 119909)is straightforward where 119891 satisfies the following growthcondition

max [1003816100381610038161003816119891 (119905 119909)1003816100381610038161003816 1003816100381610038161003816120597119905119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119909119891 (119905 119909)

1003816100381610038161003816]

le 119888119890120582|119909|2

(91)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1

Theorem 50 (Ito formula for extended Skorohod integralsII) Let 119883 be a separable centered Gaussian process withcovariance 119877 and assume that 119891 isin 11986212 satisfies growthcondition (91) Furthermore assume that (H) holds and thatthe variance of119883 is bounded and is of bounded variationThenfor any 119905 isin [0 119879] the process 120597

119909119891(sdot 119883

sdot)1

119905belongs to Dom119864

120575119879

and

119891 (119905 119883119905) = 119891 (0 119883

0) + int

119905

0

120597119909119891 (119904 119883

119904) 120575119883

119904

+ int

119905

0

120597119905119891 (119904 119883

119904) d119904

+1

2int

119905

0

120597119909119909119891 (119904 119883

119904) d119877 (119904 119904)

(92)

Proof Taking into account that we have no problems con-cerning processes to belong to the required spaces theformula follows by approximating with polynomials of form119901(119909)119902(119905) and following the proof of Theorem 40

5 Applications

We illustrate how some results transfer easily from theBrownian case to the Gaussian Fredholm processes

51 Equivalence in Law The transfer principle has alreadybeen used in connection with the equivalence of law of

Gaussian processes in for example [24] in the context offractional Brownian motions and in [2] in the context ofGaussianVolterra processes satisfying certain nondegeneracyconditions The following proposition uses the Fredholmrepresentation (8) to give a sufficient condition for theequivalence of general Gaussian processes in terms of theirFredholm kernels

Proposition 51 Let 119883 and be two Gaussian processes withFredholm kernels 119870

119879and

119879 respectively If there exists a

Volterra kernel ℓ isin 1198712([0 119879]2) such that

119879(119905 119904) = 119870

119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 (93)

then119883 and are equivalent in law

Proof Recall that by the Hitsuda representation theorem [5Theorem 63] a centered Gaussian process is equivalent inlaw to a Brownian motion on [0 119879] if and only if there existsa kernel ℓ isin 1198712([0 119879]2) and a Brownian motion119882 such that admits the representation

119905= 119882

119905minus int

119905

0

int

119904

0

ℓ (119904 119906) d119882119906d119904 (94)

Let119883 have the Fredholm representations

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (95)

Then is equivalent to119883 if it admits in law the representa-tion

119905

119889

= int

119879

0

119870119879(119905 119904) d

119904 (96)

where is connected to119882 of (95) by (94)In order to show (96) let

119905= int

119879

0

119879(119905 119904) d1198821015840

119905(97)

be the Fredholm representation of Here 1198821015840 is someBrownian motion Then by using connection (93) and theFubini theorem we obtain

119905= int

119879

0

119879(119905 119904) d1198821015840

119904

119889

= int

119879

0

119879(119905 119904) d119882

119904

= int

119879

0

(119870119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906) d119882

119904

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

International Journal of Stochastic Analysis 7

where 119891 isin 119862infin119887(R119899) that is 119891 and all its derivatives are

bounded The Malliavin derivative 119863119879119865 = 119863

119883

119879119865 of 119865 is an

element of 1198712(ΩH119879) defined by

119863119879119865 =

119899

sum

119894=1

120597119894119891 (119883 (120593

1) 119883 (120593

119899)) 120593

119894 (45)

In particular119863119879119883119905= 1

119905

Definition 32 Let D12= D12

119883be the Hilbert space of all

square integrable Malliavin differentiable random variablesdefined as the closure of S with respect to norm

1198652

12= E [|119865|

2] + E [

10038171003817100381710038171198631198791198651003817100381710038171003817

2

H119879] (46)

The divergence operator 120575119879is defined as the adjoint

operator of the Malliavin derivative119863119879

Definition 33 The domain Dom 120575119879of the operator 120575

119879is the

set of random variables 119906 isin 1198712(ΩH119879) satisfying

10038161003816100381610038161003816E ⟨119863

119879119865 119906⟩

H119879

10038161003816100381610038161003816le 119888

119906119865

1198712 (47)

for any 119865 isin D12 and some constant 119888119906depending only on 119906

For 119906 isin Dom 120575119879the divergence operator 120575

119879(119906) is a square

integrable random variable defined by the duality relation

E [119865120575119879(119906)] = E ⟨119863

119879119865 119906⟩

H119879(48)

for all 119865 isin D12

Remark 34 It is well-known that D12sub Dom 120575

119879

We use the notation

120575119879(119906) = int

119879

0

119906119904120575119883

119904 (49)

Theorem 35 (transfer principle for Malliavin calculus) Let119883 be a separable centered Gaussian process with Fredholmrepresentation (8) Let 119863

119879and 120575

119879be the Malliavin derivative

and the Skorohod integral with respect to119883 on [0 119879] Similarlylet 119863119882

119879and 120575119882

119879be the Malliavin derivative and the Skorohod

integral with respect to the Brownianmotion119882 of (8) restrictedon [0 119879] Then

120575119879= 120575

119882

119879119870lowast

119879

119870lowast

119879119863119879= 119863

119882

119879

(50)

Proof The proof follows directly from transfer principle andthe isometry provided by 119870lowast

119879with the same arguments as in

[1] Indeed by isometry we have

H119879= (119870

lowast

119879)minus1

(1198712([0 119879])) (51)

where (119870lowast

119879)minus1 denotes the preimage which implies that

D12(H

119879) = (119870

lowast

119879)minus1

(D12

119882(119871

2([0 119879]))) (52)

which justifies 119870lowast

119879119863119879= 119863

119882

119879 Furthermore we have relation

E ⟨119906119863119879119865⟩

H119879= E ⟨119870

lowast

119879119906119863

119882

119879119865⟩

1198712([0119879])

(53)

for any smooth random variable 119865 and 119906 isin 1198712(ΩH119879)

Hence by the very definition ofDom 120575 and transfer principlewe obtain

Dom 120575119879= (119870

lowast

119879)minus1

(Dom 120575119882119879) (54)

and 120575119879(119906) = 120575

119882

119879(119870

lowast

119879119906) proving the claim

Now we are ready to show that the definition of themultiple Wiener integral 119868

119879119901in Section 42 is correct in the

sense that it agrees with the iterated Skorohod integral

Proposition 36 Let ℎ isin H119879119901

be of form ℎ(1199091 119909

119901) =

prod119901

119896=1ℎ119896(119909

119896) Then ℎ is iteratively 119901 times Skorohod integrable

and

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901) 120575119883

1199051sdot sdot sdot 120575119883

119905119901= 119868

119879119901(ℎ) (55)

Moreover if ℎ isin H0

119879119901is such that it is 119901 times iteratively

Skorohod integrable then (55) still holds

Proof Again the idea is to use the transfer principle togetherwith induction Note first that the statement is true for 119901 = 1by definition and assume next that the statement is validfor 119896 = 1 119901 We denote 119891

119895= prod

119895

119896=1ℎ119896(119909

119896) Hence by

induction assumption we have

int

119879

0

sdot sdot sdot int

119879

0

ℎ (1199051 119905

119901 119905V) 1205751198831199051

sdot sdot sdot 120575119883119905119901120575119883V

= int

119879

0

119868119879119901(119891

119901) ℎ

119901+1(V) 120575119883V

(56)

Put now 119865 = 119868119879119901(119891

119901) and 119906(119905) = ℎ

119901+1(119905) Hence by [22

Proposition 133] and by applying the transfer principle weobtain that 119865119906 belongs to Dom 120575

119879and

120575119879(119865119906) = 120575

119879(119906) 119865 minus ⟨119863

119905119865 119906 (119905)⟩

H119879

= 119868119882

119879(119870

lowast

119879ℎ119901+1) 119868

119882

119879119901(119870

lowast

119879119901119891119901)

minus 119901 ⟨119868119879119901minus1(119891

119901(sdot 119905)) ℎ

119901+1(119905)⟩

H119879

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119882

119879119901minus1(119870

lowast

119879119901minus1119891119901otimes1119870lowast

119879ℎ119901+1)

= 119868119879(ℎ

119901+1) 119868

119879119901(119891

119901)

minus 119901119868119879119901minus1(119891

119901otimes1198701198791ℎ119901+1)

(57)

Hence the result is valid also for 119901+ 1 by Proposition 28 with119902 = 1

The claim for general ℎ isinH0

119879119901follows by approximating

with a product of simple function

8 International Journal of Stochastic Analysis

Remark 37 Note that (55) does not hold for arbitrary ℎ isinH0

119879119901in general without the a priori assumption of 119901 times

iterative Skorohod integrability For example let 119901 = 2 and119883 = 119861

119867 be a fractional Brownian motion with119867 le 14 anddefine ℎ

119905(119904 V) = 1

119905(119904)1

119904(V) for some fixed 119905 isin [0 119879] Then

int

119879

0

ℎ119905(119904 V) 120575119883V = 119883119904

1119905(119904) (58)

But119883sdot1119905does not belong to Dom 120575

119879(see [7])

We end this section by providing an extension of Ito for-mulas provided by Alos et al [1] They considered GaussianVolterra processes that is they assumed the representation

119883119905= int

119905

0

119870 (119905 119904) d119882119904 (59)

where the kernel119870 satisfied certain technical assumptions In[1] it was proved that in the case of Volterra processes one has

119891 (119883119905) = 119891 (0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(60)

if 119891 satisfies the growth condition

max [1003816100381610038161003816119891 (119909)1003816100381610038161003816 100381610038161003816100381610038161198911015840(119909)100381610038161003816100381610038161003816100381610038161003816100381611989110158401015840(119909)10038161003816100381610038161003816] le 119888119890

120582|119909|2

(61)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1 In the

following we will consider different approach which enablesus to

(i) prove that such formula holds with minimal require-ments

(ii) give more instructive proof of such result(iii) extend the result from Volterra context to more

general Gaussian processes(iv) drop some technical assumptions posed in [1]

For simplicity we assume that the variance of 119883 is ofbounded variation to guarantee the existence of the integral

int

119879

0

11989110158401015840(119883

119905) d119877 (119905 119905) (62)

If the variance is not of bounded variation then integral (62)may be understood by integration by parts if 11989110158401015840 is smoothenough or in the general case via the inner product ⟨sdot sdot⟩H119879 InTheorem 40 we also have to assume that the variance of119883is bounded

The result for polynomials is straightforward once weassume that the paths of polynomials of 119883 belong to1198712(ΩH

119879)

Proposition 38 (Ito formula for polynomials) Let 119883 be aseparable centered Gaussian process with covariance 119877 andassume that 119901 is a polynomial Furthermore assume that for

each polynomial 119901 one has 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) Then for

each 119905 isin [0 119879] one has

119901 (119883119905) = 119901 (119883

0) + int

119905

0

1199011015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11990110158401015840(119883

119904) d119877 (119904 119904)

(63)

if and only if119883sdot1119905belongs to Dom 120575

119879

Remark 39 The message of the above result is that once theprocesses 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) then they automatically

belong to the domain of 120575119879which is a subspace of 1198712(ΩH

119879)

However in order to check 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) one needs

more information on the kernel 119870119879 A sufficient condition is

provided in Corollary 43 which covers many cases of interest

Proof By definition and applying transfer principle we haveto prove that 1199011015840(119883

sdot)1

119905belongs to domain of 120575

119879and that

Eint119905

0

119863119904119866119870

lowast

119879[119901

1015840(119883

sdot) 1

119905] d119904

= E [119866119901 (119883119905)] minus E [119866119901 (119883

0)]

minus1

2int

119905

0

E [11986611990110158401015840(119883

119905)] d119877 (119905 119905)

(64)

for every random variable 119866 from a total subset of 1198712(Ω)In other words it is sufficient to show that (69) is valid forrandom variables of form 119866 = 119868119882

119899(ℎ

otimes119899) where ℎ is a step-

functionNote first that it is sufficient to prove the claim only for

Hermite polynomials119867119896 119896 = 1 Indeed it is well-known

that any polynomial can be expressed as a linear combinationof Hermite polynomials and consequently the result forarbitrary polynomial 119901 follows by linearity

We proceed by induction First it is clear that firsttwo polynomials 119867

0and 119867

1satisfy (64) Furthermore by

assumption 1198671015840

2(119883

sdot)1

119905belongs to Dom 120575

119879from which (64)

is easily deduced by [22 Proposition 133] Assume nextthat the result is valid for Hermite polynomials 119867

119896 119896 =

0 1 119899 Then recall well-known recursion formulas

119867119899+1(119909) = 119909119867

119899(119909) minus 119899119867

119899minus1(119909)

1198671015840

119899(119909) = 119899119867

119899minus1(119909)

(65)

The induction step follows with straightforward calcula-tions by using the recursion formulas above and [22Proposition 133] We leave the details to the reader

We will now illustrate how the result can be generalizedfor functions satisfying the growth condition (61) by usingProposition 38 First note that the growth condition (61) isindeed natural since it guarantees that the left side of (60)is square integrable Consequently since operator 120575

119879is a

mapping from 1198712(ΩH119879) into 1198712(Ω) functions satisfying

(61) are the largest class of functions for which (60) can holdHowever it is not clear in general whether 1198911015840(119883

sdot)1

119905belongs

International Journal of Stochastic Analysis 9

to Dom 120575119879 Indeed for example in [1] the authors posed

additional conditions on the Volterra kernel 119870 to guaranteethis As our main result we show that E1198911015840(119883

sdot)1

1199052

H119879lt infin

implies that (60) holds In other words not only is the Itoformula (60) natural but it is also the only possibility

Theorem 40 (Ito formula for Skorohod integrals) Let 119883 bea separable centered Gaussian process with covariance 119877 suchthat all the polynomials 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) Assume that

119891 isin 1198622 satisfies growth condition (61) and that the variance of

119883 is bounded and of bounded variation If

E100381710038171003817100381710038171198911015840(119883

sdot) 1

119905

10038171003817100381710038171003817

2

H119879lt infin (66)

for any 119905 isin [0 119879] then

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(67)

Proof In this proof we assume for notational simplicity andwith no loss of generality that sup

0le119904le119879119877(119904 119904) = 1

First it is clear that (66) implies that 1198911015840(119883sdot)1

119905belongs to

domain of 120575119879 Hence we only have to prove that

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E [119866119891 (119883119905)] minus E [119866119891 (119883

0)]

minus1

2int

119905

0

E [11986611989110158401015840(119883

119904)] d119877 (119904 119904)

(68)

for every random variable 119866 = 119868119882119899(ℎ

otimes119899)

Now it is well-known that Hermite polynomials whenproperly scaled form an orthogonal system in 1198712(R) whenequipped with the Gaussian measure Now each 119891 satisfyingthe growth condition (61) has a series representation

119891 (119909) =

infin

sum

119896=0

120572119896119867119896(119909) (69)

Indeed the growth condition (61) implies that

intR

100381610038161003816100381610038161198911015840(119909)10038161003816100381610038161003816

2

119890minus1199092(2sup

0le119904le119879119877(119904119904))d119909 lt infin (70)

Furthermore we have

119891 (119883119904) =

infin

sum

119896=0

120572119896119867119896(119883

119904) (71)

where the series converge almost surely and in 1198712(Ω) andsimilar conclusion is valid for derivatives1198911015840(119883

119904) and 11989110158401015840(119883

119904)

Then by applying (66) we obtain that for any 120598 gt 0 thereexists119873 = 119873

120598such that we have

E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598 119899 ge 119873 (72)

where

1198911015840

119899(119883

119904) =

infin

sum

119896=119899

120572119896119867

1015840

119896(119883

119904) (73)

Consequently for random variables of form 119866 = 119868119882119899(ℎ

otimes119899) we

obtain by choosing 119873 large enough and applying Proposi-tion 38 that

E [119866119891 (119883119905)] minus E [119866119891 (119883

0)] minus1

2

sdot int

119905

0

E (11986611989110158401015840(119883

119905)) d119877 (119905 119905)

minus E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598

(74)

Now the left side does not depend on 119899 which concludes theproof

Remark 41 Note that actually it is sufficient to have

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

=

infin

sum

119896=1

120572119896E ⟨119863

119879119866119867

1015840

119896(119883

sdot) 1

119905⟩H119879

(75)

from which the result follows by Proposition 38 Further-more taking account growth condition (61) this is actuallysufficient and necessary condition for formula (60) to holdConsequently our method can also be used to obtain Itoformulas by considering extended domain of 120575

119879(see [9] or

[11]) This is the topic of Section 44

Example 42 It is known that if 119883 = 119861119867 is a fractional Brow-nian motion with119867 gt 14 then 1198911015840(119883

sdot)1

119905satisfies condition

(66) while for 119867 le 14 it does not (see [22 Chapter 5])Consequently a simple application of Theorem 40 coversfractional Brownian motion with 119867 gt 14 For the case119867 le 14 one has to consider extended domain of 120575

119879which is

proved in [9] Consequently in this case we have (75) for any119865 isin S

We end this section by illustrating the power of ourmethod with the following simple corollary which is anextension of [1 Theorem 1]

Corollary 43 Let 119883 be a separable centered continuousGaussian process with covariance 119877 that is bounded such thatthe Fredholm kernel 119870

119879is of bounded variation and

int

119879

0

(int

119879

0

1003817100381710038171003817119883119905minus 119883

119904

10038171003817100381710038171198712(Ω)

10038161003816100381610038161198701198791003816100381610038161003816 (d119905 119904))

2

d119904 lt infin (76)

Then for any 119905 isin [0 119879] one has

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(77)

10 International Journal of Stochastic Analysis

Proof Note that assumption is a Fredholm version of con-dition (K2) in [1] which implies condition (66) Hence theresult follows byTheorem 40

44 Extended Divergence Operator As shown in Section 43the Ito formula (60) is the only possibility However theproblem is that the space 1198712(ΩH

119879) may be too small to

contain the elements 1198911015840(119883sdot)1

119905 In particular it may happen

that not even the process 119883 itself belongs to 1198712(ΩH119879) (see

eg [7] for the case of fractional Brownian motion with119867 le 14) This problem can be overcome by consideringan extended domain of 120575

119879 The idea of extended domain is

to extend the inner product ⟨119906 120593⟩H119879 for simple 120593 to moregeneral processes 119906 and then define extended domain by(47) with a restricted class of test variables 119865 This also givesanother intuitive reason why extended domain of 120575

119879can be

useful indeed here we have proved that Ito formula (60) isthe only possibility and what one essentially needs for suchresult is the following

(i) 119883sdot1119905belongs to Dom 120575

119879

(ii) Equation (75) is valid for functions satisfying (61)

Consequently one should look for extensions of operator 120575119879

such that these two things are satisfiedTo facilitate the extension of domain we make the

following relatively moderate assumption

(H) The function 119905 997891rarr 119877(119905 119904) is of bounded variation on[0 119879] and

sup119905isin[0119879]

int

119879

0

|119877| (d119904 119905) lt infin (78)

Remark 44 Note that we are making the assumption on thecovariance 119877 not the kernel 119870

119879 Hence our case is different

from that of [1] Also [11] assumed absolute continuity in 119877we are satisfied with bounded variation

We will follow the idea from Lei and Nualart [11] andextend the inner product ⟨sdot sdot⟩H119879 beyondH

119879

Consider a step-function 120593Then on the one hand by theisometry property we have

⟨120593 1119905⟩H119879= int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 (79)

where 119892119905(119904) = 119870(119905 119904) isin 119871

2([0 119879]) On the other hand by

using adjoint property (see Remark 20) we obtain

int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 = int

119879

0

120593 (119904) (119870119879119892119905) (d119904) (80)

where computing formally we have

(119870119879119892119905) (d119904) = int

119879

0

119892119905(119906)119870

119879(d119904 119906) d119906

= int

119879

0

119870119879(119905 119906)119870

119879(d119904 119906) d119906 = 119877 (119905 d119904)

(81)

Consequently

⟨120593 1119905⟩H119879= int

119879

0

120593 (119904) 119877 (119905 d119904) (82)

This gives motivation to the following definition similar tothat of [11 Definition 21]

Definition 45 Denote by T119879the space of measurable func-

tions 119892 satisfying

sup119905isin[0119879]

int

119879

0

1003816100381610038161003816119892 (119904)1003816100381610038161003816 |119877| (119905 d119904) lt infin (83)

and let 120593 be a step-function of form 120593 = sum119899119896=11198871198961119905119896 Then we

extend ⟨sdot sdot⟩H119879 toT119879by defining

⟨119892 120593⟩H119879=

119899

sum

119896=1

119887119896int

119879

0

119892 (119904) 119877 (119905119896 d119904) (84)

In particular this implies that for 119892 and 120593 as above we have

⟨1198921119905 120593⟩

H119879= int

119905

0

119892 (119904) d ⟨1119904 120593⟩

H119879 (85)

We define extended domain Dom119864120575119879similarly as in [11]

Definition 46 A process 119906 isin 1198711(ΩT119879) belongs to Dom119864

120575119879

if10038161003816100381610038161003816E ⟨119906119863119865⟩H119879

10038161003816100381610038161003816le 119888

119906119865

2(86)

for any smooth random variable 119865 isin S In this case 120575(119906) isin1198712(Ω) is defined by duality relationship

E [119865120575 (119906)] = E ⟨119906119863119865⟩H119879 (87)

Remark 47 Note that in general Dom 120575119879and Dom119864

120575119879are

not comparable See [11] for discussion

Note now that if a function 119891 satisfies the growthcondition (61) then 1198911015840(119883

sdot)1

119905isin 119871

1(ΩT

119879) since (61) implies

E sup119905isin[0119879]

100381610038161003816100381610038161198911015840(119883

119905)10038161003816100381610038161003816

119901

lt infin (88)

for any 119901 lt (12120582)(sup119905isin[0119879]

119877(119905 119905))minus1 Consequently with

this definition we are able to get rid of the problem thatprocesses might not belong to corresponding H

119879-spaces

Furthermore this implies that the series expansion (69)converges in the norm 1198711(ΩT

119879) defined by

Eint119879

0

|119906 (119904)| |119877| (119905 d119904) (89)

which in turn implies (75) Hence it is straightforward toobtain the following by first showing the result for polynomi-als and then by approximating in a similar manner as done inSection 43 but using the extended domain instead

International Journal of Stochastic Analysis 11

Theorem 48 (Ito formula for extended Skorohod integrals)Let119883 be a separable centered Gaussian process with covariance119877 and assume that 119891 isin 1198622 satisfies growth condition (61)Furthermore assume that (H) holds and that the variance of119883is bounded and is of bounded variationThen for any 119905 isin [0 119879]the process 1198911015840(119883

sdot)1

119905belongs to Dom119864

120575119879and

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(90)

Remark 49 As an application of Theorem 48 it is straight-forward to derive version of Ito-Tanaka formula underadditional conditions which guarantee that for a certainsequence of functions 119891

119899we have the convergence of term

(12) int119905

011989110158401015840

119899(119883

119904)d119877(119904 119904) to the local time For details we refer

to [11] where authors derived such formula under theirassumptions

Finally let us note that the extension to functions 119891(119905 119909)is straightforward where 119891 satisfies the following growthcondition

max [1003816100381610038161003816119891 (119905 119909)1003816100381610038161003816 1003816100381610038161003816120597119905119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119909119891 (119905 119909)

1003816100381610038161003816]

le 119888119890120582|119909|2

(91)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1

Theorem 50 (Ito formula for extended Skorohod integralsII) Let 119883 be a separable centered Gaussian process withcovariance 119877 and assume that 119891 isin 11986212 satisfies growthcondition (91) Furthermore assume that (H) holds and thatthe variance of119883 is bounded and is of bounded variationThenfor any 119905 isin [0 119879] the process 120597

119909119891(sdot 119883

sdot)1

119905belongs to Dom119864

120575119879

and

119891 (119905 119883119905) = 119891 (0 119883

0) + int

119905

0

120597119909119891 (119904 119883

119904) 120575119883

119904

+ int

119905

0

120597119905119891 (119904 119883

119904) d119904

+1

2int

119905

0

120597119909119909119891 (119904 119883

119904) d119877 (119904 119904)

(92)

Proof Taking into account that we have no problems con-cerning processes to belong to the required spaces theformula follows by approximating with polynomials of form119901(119909)119902(119905) and following the proof of Theorem 40

5 Applications

We illustrate how some results transfer easily from theBrownian case to the Gaussian Fredholm processes

51 Equivalence in Law The transfer principle has alreadybeen used in connection with the equivalence of law of

Gaussian processes in for example [24] in the context offractional Brownian motions and in [2] in the context ofGaussianVolterra processes satisfying certain nondegeneracyconditions The following proposition uses the Fredholmrepresentation (8) to give a sufficient condition for theequivalence of general Gaussian processes in terms of theirFredholm kernels

Proposition 51 Let 119883 and be two Gaussian processes withFredholm kernels 119870

119879and

119879 respectively If there exists a

Volterra kernel ℓ isin 1198712([0 119879]2) such that

119879(119905 119904) = 119870

119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 (93)

then119883 and are equivalent in law

Proof Recall that by the Hitsuda representation theorem [5Theorem 63] a centered Gaussian process is equivalent inlaw to a Brownian motion on [0 119879] if and only if there existsa kernel ℓ isin 1198712([0 119879]2) and a Brownian motion119882 such that admits the representation

119905= 119882

119905minus int

119905

0

int

119904

0

ℓ (119904 119906) d119882119906d119904 (94)

Let119883 have the Fredholm representations

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (95)

Then is equivalent to119883 if it admits in law the representa-tion

119905

119889

= int

119879

0

119870119879(119905 119904) d

119904 (96)

where is connected to119882 of (95) by (94)In order to show (96) let

119905= int

119879

0

119879(119905 119904) d1198821015840

119905(97)

be the Fredholm representation of Here 1198821015840 is someBrownian motion Then by using connection (93) and theFubini theorem we obtain

119905= int

119879

0

119879(119905 119904) d1198821015840

119904

119889

= int

119879

0

119879(119905 119904) d119882

119904

= int

119879

0

(119870119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906) d119882

119904

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

8 International Journal of Stochastic Analysis

Remark 37 Note that (55) does not hold for arbitrary ℎ isinH0

119879119901in general without the a priori assumption of 119901 times

iterative Skorohod integrability For example let 119901 = 2 and119883 = 119861

119867 be a fractional Brownian motion with119867 le 14 anddefine ℎ

119905(119904 V) = 1

119905(119904)1

119904(V) for some fixed 119905 isin [0 119879] Then

int

119879

0

ℎ119905(119904 V) 120575119883V = 119883119904

1119905(119904) (58)

But119883sdot1119905does not belong to Dom 120575

119879(see [7])

We end this section by providing an extension of Ito for-mulas provided by Alos et al [1] They considered GaussianVolterra processes that is they assumed the representation

119883119905= int

119905

0

119870 (119905 119904) d119882119904 (59)

where the kernel119870 satisfied certain technical assumptions In[1] it was proved that in the case of Volterra processes one has

119891 (119883119905) = 119891 (0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(60)

if 119891 satisfies the growth condition

max [1003816100381610038161003816119891 (119909)1003816100381610038161003816 100381610038161003816100381610038161198911015840(119909)100381610038161003816100381610038161003816100381610038161003816100381611989110158401015840(119909)10038161003816100381610038161003816] le 119888119890

120582|119909|2

(61)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1 In the

following we will consider different approach which enablesus to

(i) prove that such formula holds with minimal require-ments

(ii) give more instructive proof of such result(iii) extend the result from Volterra context to more

general Gaussian processes(iv) drop some technical assumptions posed in [1]

For simplicity we assume that the variance of 119883 is ofbounded variation to guarantee the existence of the integral

int

119879

0

11989110158401015840(119883

119905) d119877 (119905 119905) (62)

If the variance is not of bounded variation then integral (62)may be understood by integration by parts if 11989110158401015840 is smoothenough or in the general case via the inner product ⟨sdot sdot⟩H119879 InTheorem 40 we also have to assume that the variance of119883is bounded

The result for polynomials is straightforward once weassume that the paths of polynomials of 119883 belong to1198712(ΩH

119879)

Proposition 38 (Ito formula for polynomials) Let 119883 be aseparable centered Gaussian process with covariance 119877 andassume that 119901 is a polynomial Furthermore assume that for

each polynomial 119901 one has 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) Then for

each 119905 isin [0 119879] one has

119901 (119883119905) = 119901 (119883

0) + int

119905

0

1199011015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11990110158401015840(119883

119904) d119877 (119904 119904)

(63)

if and only if119883sdot1119905belongs to Dom 120575

119879

Remark 39 The message of the above result is that once theprocesses 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) then they automatically

belong to the domain of 120575119879which is a subspace of 1198712(ΩH

119879)

However in order to check 119901(119883sdot)1

119905isin 119871

2(ΩH

119879) one needs

more information on the kernel 119870119879 A sufficient condition is

provided in Corollary 43 which covers many cases of interest

Proof By definition and applying transfer principle we haveto prove that 1199011015840(119883

sdot)1

119905belongs to domain of 120575

119879and that

Eint119905

0

119863119904119866119870

lowast

119879[119901

1015840(119883

sdot) 1

119905] d119904

= E [119866119901 (119883119905)] minus E [119866119901 (119883

0)]

minus1

2int

119905

0

E [11986611990110158401015840(119883

119905)] d119877 (119905 119905)

(64)

for every random variable 119866 from a total subset of 1198712(Ω)In other words it is sufficient to show that (69) is valid forrandom variables of form 119866 = 119868119882

119899(ℎ

otimes119899) where ℎ is a step-

functionNote first that it is sufficient to prove the claim only for

Hermite polynomials119867119896 119896 = 1 Indeed it is well-known

that any polynomial can be expressed as a linear combinationof Hermite polynomials and consequently the result forarbitrary polynomial 119901 follows by linearity

We proceed by induction First it is clear that firsttwo polynomials 119867

0and 119867

1satisfy (64) Furthermore by

assumption 1198671015840

2(119883

sdot)1

119905belongs to Dom 120575

119879from which (64)

is easily deduced by [22 Proposition 133] Assume nextthat the result is valid for Hermite polynomials 119867

119896 119896 =

0 1 119899 Then recall well-known recursion formulas

119867119899+1(119909) = 119909119867

119899(119909) minus 119899119867

119899minus1(119909)

1198671015840

119899(119909) = 119899119867

119899minus1(119909)

(65)

The induction step follows with straightforward calcula-tions by using the recursion formulas above and [22Proposition 133] We leave the details to the reader

We will now illustrate how the result can be generalizedfor functions satisfying the growth condition (61) by usingProposition 38 First note that the growth condition (61) isindeed natural since it guarantees that the left side of (60)is square integrable Consequently since operator 120575

119879is a

mapping from 1198712(ΩH119879) into 1198712(Ω) functions satisfying

(61) are the largest class of functions for which (60) can holdHowever it is not clear in general whether 1198911015840(119883

sdot)1

119905belongs

International Journal of Stochastic Analysis 9

to Dom 120575119879 Indeed for example in [1] the authors posed

additional conditions on the Volterra kernel 119870 to guaranteethis As our main result we show that E1198911015840(119883

sdot)1

1199052

H119879lt infin

implies that (60) holds In other words not only is the Itoformula (60) natural but it is also the only possibility

Theorem 40 (Ito formula for Skorohod integrals) Let 119883 bea separable centered Gaussian process with covariance 119877 suchthat all the polynomials 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) Assume that

119891 isin 1198622 satisfies growth condition (61) and that the variance of

119883 is bounded and of bounded variation If

E100381710038171003817100381710038171198911015840(119883

sdot) 1

119905

10038171003817100381710038171003817

2

H119879lt infin (66)

for any 119905 isin [0 119879] then

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(67)

Proof In this proof we assume for notational simplicity andwith no loss of generality that sup

0le119904le119879119877(119904 119904) = 1

First it is clear that (66) implies that 1198911015840(119883sdot)1

119905belongs to

domain of 120575119879 Hence we only have to prove that

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E [119866119891 (119883119905)] minus E [119866119891 (119883

0)]

minus1

2int

119905

0

E [11986611989110158401015840(119883

119904)] d119877 (119904 119904)

(68)

for every random variable 119866 = 119868119882119899(ℎ

otimes119899)

Now it is well-known that Hermite polynomials whenproperly scaled form an orthogonal system in 1198712(R) whenequipped with the Gaussian measure Now each 119891 satisfyingthe growth condition (61) has a series representation

119891 (119909) =

infin

sum

119896=0

120572119896119867119896(119909) (69)

Indeed the growth condition (61) implies that

intR

100381610038161003816100381610038161198911015840(119909)10038161003816100381610038161003816

2

119890minus1199092(2sup

0le119904le119879119877(119904119904))d119909 lt infin (70)

Furthermore we have

119891 (119883119904) =

infin

sum

119896=0

120572119896119867119896(119883

119904) (71)

where the series converge almost surely and in 1198712(Ω) andsimilar conclusion is valid for derivatives1198911015840(119883

119904) and 11989110158401015840(119883

119904)

Then by applying (66) we obtain that for any 120598 gt 0 thereexists119873 = 119873

120598such that we have

E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598 119899 ge 119873 (72)

where

1198911015840

119899(119883

119904) =

infin

sum

119896=119899

120572119896119867

1015840

119896(119883

119904) (73)

Consequently for random variables of form 119866 = 119868119882119899(ℎ

otimes119899) we

obtain by choosing 119873 large enough and applying Proposi-tion 38 that

E [119866119891 (119883119905)] minus E [119866119891 (119883

0)] minus1

2

sdot int

119905

0

E (11986611989110158401015840(119883

119905)) d119877 (119905 119905)

minus E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598

(74)

Now the left side does not depend on 119899 which concludes theproof

Remark 41 Note that actually it is sufficient to have

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

=

infin

sum

119896=1

120572119896E ⟨119863

119879119866119867

1015840

119896(119883

sdot) 1

119905⟩H119879

(75)

from which the result follows by Proposition 38 Further-more taking account growth condition (61) this is actuallysufficient and necessary condition for formula (60) to holdConsequently our method can also be used to obtain Itoformulas by considering extended domain of 120575

119879(see [9] or

[11]) This is the topic of Section 44

Example 42 It is known that if 119883 = 119861119867 is a fractional Brow-nian motion with119867 gt 14 then 1198911015840(119883

sdot)1

119905satisfies condition

(66) while for 119867 le 14 it does not (see [22 Chapter 5])Consequently a simple application of Theorem 40 coversfractional Brownian motion with 119867 gt 14 For the case119867 le 14 one has to consider extended domain of 120575

119879which is

proved in [9] Consequently in this case we have (75) for any119865 isin S

We end this section by illustrating the power of ourmethod with the following simple corollary which is anextension of [1 Theorem 1]

Corollary 43 Let 119883 be a separable centered continuousGaussian process with covariance 119877 that is bounded such thatthe Fredholm kernel 119870

119879is of bounded variation and

int

119879

0

(int

119879

0

1003817100381710038171003817119883119905minus 119883

119904

10038171003817100381710038171198712(Ω)

10038161003816100381610038161198701198791003816100381610038161003816 (d119905 119904))

2

d119904 lt infin (76)

Then for any 119905 isin [0 119879] one has

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(77)

10 International Journal of Stochastic Analysis

Proof Note that assumption is a Fredholm version of con-dition (K2) in [1] which implies condition (66) Hence theresult follows byTheorem 40

44 Extended Divergence Operator As shown in Section 43the Ito formula (60) is the only possibility However theproblem is that the space 1198712(ΩH

119879) may be too small to

contain the elements 1198911015840(119883sdot)1

119905 In particular it may happen

that not even the process 119883 itself belongs to 1198712(ΩH119879) (see

eg [7] for the case of fractional Brownian motion with119867 le 14) This problem can be overcome by consideringan extended domain of 120575

119879 The idea of extended domain is

to extend the inner product ⟨119906 120593⟩H119879 for simple 120593 to moregeneral processes 119906 and then define extended domain by(47) with a restricted class of test variables 119865 This also givesanother intuitive reason why extended domain of 120575

119879can be

useful indeed here we have proved that Ito formula (60) isthe only possibility and what one essentially needs for suchresult is the following

(i) 119883sdot1119905belongs to Dom 120575

119879

(ii) Equation (75) is valid for functions satisfying (61)

Consequently one should look for extensions of operator 120575119879

such that these two things are satisfiedTo facilitate the extension of domain we make the

following relatively moderate assumption

(H) The function 119905 997891rarr 119877(119905 119904) is of bounded variation on[0 119879] and

sup119905isin[0119879]

int

119879

0

|119877| (d119904 119905) lt infin (78)

Remark 44 Note that we are making the assumption on thecovariance 119877 not the kernel 119870

119879 Hence our case is different

from that of [1] Also [11] assumed absolute continuity in 119877we are satisfied with bounded variation

We will follow the idea from Lei and Nualart [11] andextend the inner product ⟨sdot sdot⟩H119879 beyondH

119879

Consider a step-function 120593Then on the one hand by theisometry property we have

⟨120593 1119905⟩H119879= int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 (79)

where 119892119905(119904) = 119870(119905 119904) isin 119871

2([0 119879]) On the other hand by

using adjoint property (see Remark 20) we obtain

int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 = int

119879

0

120593 (119904) (119870119879119892119905) (d119904) (80)

where computing formally we have

(119870119879119892119905) (d119904) = int

119879

0

119892119905(119906)119870

119879(d119904 119906) d119906

= int

119879

0

119870119879(119905 119906)119870

119879(d119904 119906) d119906 = 119877 (119905 d119904)

(81)

Consequently

⟨120593 1119905⟩H119879= int

119879

0

120593 (119904) 119877 (119905 d119904) (82)

This gives motivation to the following definition similar tothat of [11 Definition 21]

Definition 45 Denote by T119879the space of measurable func-

tions 119892 satisfying

sup119905isin[0119879]

int

119879

0

1003816100381610038161003816119892 (119904)1003816100381610038161003816 |119877| (119905 d119904) lt infin (83)

and let 120593 be a step-function of form 120593 = sum119899119896=11198871198961119905119896 Then we

extend ⟨sdot sdot⟩H119879 toT119879by defining

⟨119892 120593⟩H119879=

119899

sum

119896=1

119887119896int

119879

0

119892 (119904) 119877 (119905119896 d119904) (84)

In particular this implies that for 119892 and 120593 as above we have

⟨1198921119905 120593⟩

H119879= int

119905

0

119892 (119904) d ⟨1119904 120593⟩

H119879 (85)

We define extended domain Dom119864120575119879similarly as in [11]

Definition 46 A process 119906 isin 1198711(ΩT119879) belongs to Dom119864

120575119879

if10038161003816100381610038161003816E ⟨119906119863119865⟩H119879

10038161003816100381610038161003816le 119888

119906119865

2(86)

for any smooth random variable 119865 isin S In this case 120575(119906) isin1198712(Ω) is defined by duality relationship

E [119865120575 (119906)] = E ⟨119906119863119865⟩H119879 (87)

Remark 47 Note that in general Dom 120575119879and Dom119864

120575119879are

not comparable See [11] for discussion

Note now that if a function 119891 satisfies the growthcondition (61) then 1198911015840(119883

sdot)1

119905isin 119871

1(ΩT

119879) since (61) implies

E sup119905isin[0119879]

100381610038161003816100381610038161198911015840(119883

119905)10038161003816100381610038161003816

119901

lt infin (88)

for any 119901 lt (12120582)(sup119905isin[0119879]

119877(119905 119905))minus1 Consequently with

this definition we are able to get rid of the problem thatprocesses might not belong to corresponding H

119879-spaces

Furthermore this implies that the series expansion (69)converges in the norm 1198711(ΩT

119879) defined by

Eint119879

0

|119906 (119904)| |119877| (119905 d119904) (89)

which in turn implies (75) Hence it is straightforward toobtain the following by first showing the result for polynomi-als and then by approximating in a similar manner as done inSection 43 but using the extended domain instead

International Journal of Stochastic Analysis 11

Theorem 48 (Ito formula for extended Skorohod integrals)Let119883 be a separable centered Gaussian process with covariance119877 and assume that 119891 isin 1198622 satisfies growth condition (61)Furthermore assume that (H) holds and that the variance of119883is bounded and is of bounded variationThen for any 119905 isin [0 119879]the process 1198911015840(119883

sdot)1

119905belongs to Dom119864

120575119879and

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(90)

Remark 49 As an application of Theorem 48 it is straight-forward to derive version of Ito-Tanaka formula underadditional conditions which guarantee that for a certainsequence of functions 119891

119899we have the convergence of term

(12) int119905

011989110158401015840

119899(119883

119904)d119877(119904 119904) to the local time For details we refer

to [11] where authors derived such formula under theirassumptions

Finally let us note that the extension to functions 119891(119905 119909)is straightforward where 119891 satisfies the following growthcondition

max [1003816100381610038161003816119891 (119905 119909)1003816100381610038161003816 1003816100381610038161003816120597119905119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119909119891 (119905 119909)

1003816100381610038161003816]

le 119888119890120582|119909|2

(91)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1

Theorem 50 (Ito formula for extended Skorohod integralsII) Let 119883 be a separable centered Gaussian process withcovariance 119877 and assume that 119891 isin 11986212 satisfies growthcondition (91) Furthermore assume that (H) holds and thatthe variance of119883 is bounded and is of bounded variationThenfor any 119905 isin [0 119879] the process 120597

119909119891(sdot 119883

sdot)1

119905belongs to Dom119864

120575119879

and

119891 (119905 119883119905) = 119891 (0 119883

0) + int

119905

0

120597119909119891 (119904 119883

119904) 120575119883

119904

+ int

119905

0

120597119905119891 (119904 119883

119904) d119904

+1

2int

119905

0

120597119909119909119891 (119904 119883

119904) d119877 (119904 119904)

(92)

Proof Taking into account that we have no problems con-cerning processes to belong to the required spaces theformula follows by approximating with polynomials of form119901(119909)119902(119905) and following the proof of Theorem 40

5 Applications

We illustrate how some results transfer easily from theBrownian case to the Gaussian Fredholm processes

51 Equivalence in Law The transfer principle has alreadybeen used in connection with the equivalence of law of

Gaussian processes in for example [24] in the context offractional Brownian motions and in [2] in the context ofGaussianVolterra processes satisfying certain nondegeneracyconditions The following proposition uses the Fredholmrepresentation (8) to give a sufficient condition for theequivalence of general Gaussian processes in terms of theirFredholm kernels

Proposition 51 Let 119883 and be two Gaussian processes withFredholm kernels 119870

119879and

119879 respectively If there exists a

Volterra kernel ℓ isin 1198712([0 119879]2) such that

119879(119905 119904) = 119870

119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 (93)

then119883 and are equivalent in law

Proof Recall that by the Hitsuda representation theorem [5Theorem 63] a centered Gaussian process is equivalent inlaw to a Brownian motion on [0 119879] if and only if there existsa kernel ℓ isin 1198712([0 119879]2) and a Brownian motion119882 such that admits the representation

119905= 119882

119905minus int

119905

0

int

119904

0

ℓ (119904 119906) d119882119906d119904 (94)

Let119883 have the Fredholm representations

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (95)

Then is equivalent to119883 if it admits in law the representa-tion

119905

119889

= int

119879

0

119870119879(119905 119904) d

119904 (96)

where is connected to119882 of (95) by (94)In order to show (96) let

119905= int

119879

0

119879(119905 119904) d1198821015840

119905(97)

be the Fredholm representation of Here 1198821015840 is someBrownian motion Then by using connection (93) and theFubini theorem we obtain

119905= int

119879

0

119879(119905 119904) d1198821015840

119904

119889

= int

119879

0

119879(119905 119904) d119882

119904

= int

119879

0

(119870119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906) d119882

119904

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

International Journal of Stochastic Analysis 9

to Dom 120575119879 Indeed for example in [1] the authors posed

additional conditions on the Volterra kernel 119870 to guaranteethis As our main result we show that E1198911015840(119883

sdot)1

1199052

H119879lt infin

implies that (60) holds In other words not only is the Itoformula (60) natural but it is also the only possibility

Theorem 40 (Ito formula for Skorohod integrals) Let 119883 bea separable centered Gaussian process with covariance 119877 suchthat all the polynomials 119901(119883

sdot)1

119905isin 119871

2(ΩH

119879) Assume that

119891 isin 1198622 satisfies growth condition (61) and that the variance of

119883 is bounded and of bounded variation If

E100381710038171003817100381710038171198911015840(119883

sdot) 1

119905

10038171003817100381710038171003817

2

H119879lt infin (66)

for any 119905 isin [0 119879] then

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(67)

Proof In this proof we assume for notational simplicity andwith no loss of generality that sup

0le119904le119879119877(119904 119904) = 1

First it is clear that (66) implies that 1198911015840(119883sdot)1

119905belongs to

domain of 120575119879 Hence we only have to prove that

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E [119866119891 (119883119905)] minus E [119866119891 (119883

0)]

minus1

2int

119905

0

E [11986611989110158401015840(119883

119904)] d119877 (119904 119904)

(68)

for every random variable 119866 = 119868119882119899(ℎ

otimes119899)

Now it is well-known that Hermite polynomials whenproperly scaled form an orthogonal system in 1198712(R) whenequipped with the Gaussian measure Now each 119891 satisfyingthe growth condition (61) has a series representation

119891 (119909) =

infin

sum

119896=0

120572119896119867119896(119909) (69)

Indeed the growth condition (61) implies that

intR

100381610038161003816100381610038161198911015840(119909)10038161003816100381610038161003816

2

119890minus1199092(2sup

0le119904le119879119877(119904119904))d119909 lt infin (70)

Furthermore we have

119891 (119883119904) =

infin

sum

119896=0

120572119896119867119896(119883

119904) (71)

where the series converge almost surely and in 1198712(Ω) andsimilar conclusion is valid for derivatives1198911015840(119883

119904) and 11989110158401015840(119883

119904)

Then by applying (66) we obtain that for any 120598 gt 0 thereexists119873 = 119873

120598such that we have

E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598 119899 ge 119873 (72)

where

1198911015840

119899(119883

119904) =

infin

sum

119896=119899

120572119896119867

1015840

119896(119883

119904) (73)

Consequently for random variables of form 119866 = 119868119882119899(ℎ

otimes119899) we

obtain by choosing 119873 large enough and applying Proposi-tion 38 that

E [119866119891 (119883119905)] minus E [119866119891 (119883

0)] minus1

2

sdot int

119905

0

E (11986611989110158401015840(119883

119905)) d119877 (119905 119905)

minus E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

= E ⟨119863119879119866 119891

1015840

119899(119883

sdot) 1

119905⟩H119879lt 120598

(74)

Now the left side does not depend on 119899 which concludes theproof

Remark 41 Note that actually it is sufficient to have

E ⟨119863119879119866 119891

1015840(119883

sdot) 1

119905⟩H119879

=

infin

sum

119896=1

120572119896E ⟨119863

119879119866119867

1015840

119896(119883

sdot) 1

119905⟩H119879

(75)

from which the result follows by Proposition 38 Further-more taking account growth condition (61) this is actuallysufficient and necessary condition for formula (60) to holdConsequently our method can also be used to obtain Itoformulas by considering extended domain of 120575

119879(see [9] or

[11]) This is the topic of Section 44

Example 42 It is known that if 119883 = 119861119867 is a fractional Brow-nian motion with119867 gt 14 then 1198911015840(119883

sdot)1

119905satisfies condition

(66) while for 119867 le 14 it does not (see [22 Chapter 5])Consequently a simple application of Theorem 40 coversfractional Brownian motion with 119867 gt 14 For the case119867 le 14 one has to consider extended domain of 120575

119879which is

proved in [9] Consequently in this case we have (75) for any119865 isin S

We end this section by illustrating the power of ourmethod with the following simple corollary which is anextension of [1 Theorem 1]

Corollary 43 Let 119883 be a separable centered continuousGaussian process with covariance 119877 that is bounded such thatthe Fredholm kernel 119870

119879is of bounded variation and

int

119879

0

(int

119879

0

1003817100381710038171003817119883119905minus 119883

119904

10038171003817100381710038171198712(Ω)

10038161003816100381610038161198701198791003816100381610038161003816 (d119905 119904))

2

d119904 lt infin (76)

Then for any 119905 isin [0 119879] one has

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(77)

10 International Journal of Stochastic Analysis

Proof Note that assumption is a Fredholm version of con-dition (K2) in [1] which implies condition (66) Hence theresult follows byTheorem 40

44 Extended Divergence Operator As shown in Section 43the Ito formula (60) is the only possibility However theproblem is that the space 1198712(ΩH

119879) may be too small to

contain the elements 1198911015840(119883sdot)1

119905 In particular it may happen

that not even the process 119883 itself belongs to 1198712(ΩH119879) (see

eg [7] for the case of fractional Brownian motion with119867 le 14) This problem can be overcome by consideringan extended domain of 120575

119879 The idea of extended domain is

to extend the inner product ⟨119906 120593⟩H119879 for simple 120593 to moregeneral processes 119906 and then define extended domain by(47) with a restricted class of test variables 119865 This also givesanother intuitive reason why extended domain of 120575

119879can be

useful indeed here we have proved that Ito formula (60) isthe only possibility and what one essentially needs for suchresult is the following

(i) 119883sdot1119905belongs to Dom 120575

119879

(ii) Equation (75) is valid for functions satisfying (61)

Consequently one should look for extensions of operator 120575119879

such that these two things are satisfiedTo facilitate the extension of domain we make the

following relatively moderate assumption

(H) The function 119905 997891rarr 119877(119905 119904) is of bounded variation on[0 119879] and

sup119905isin[0119879]

int

119879

0

|119877| (d119904 119905) lt infin (78)

Remark 44 Note that we are making the assumption on thecovariance 119877 not the kernel 119870

119879 Hence our case is different

from that of [1] Also [11] assumed absolute continuity in 119877we are satisfied with bounded variation

We will follow the idea from Lei and Nualart [11] andextend the inner product ⟨sdot sdot⟩H119879 beyondH

119879

Consider a step-function 120593Then on the one hand by theisometry property we have

⟨120593 1119905⟩H119879= int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 (79)

where 119892119905(119904) = 119870(119905 119904) isin 119871

2([0 119879]) On the other hand by

using adjoint property (see Remark 20) we obtain

int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 = int

119879

0

120593 (119904) (119870119879119892119905) (d119904) (80)

where computing formally we have

(119870119879119892119905) (d119904) = int

119879

0

119892119905(119906)119870

119879(d119904 119906) d119906

= int

119879

0

119870119879(119905 119906)119870

119879(d119904 119906) d119906 = 119877 (119905 d119904)

(81)

Consequently

⟨120593 1119905⟩H119879= int

119879

0

120593 (119904) 119877 (119905 d119904) (82)

This gives motivation to the following definition similar tothat of [11 Definition 21]

Definition 45 Denote by T119879the space of measurable func-

tions 119892 satisfying

sup119905isin[0119879]

int

119879

0

1003816100381610038161003816119892 (119904)1003816100381610038161003816 |119877| (119905 d119904) lt infin (83)

and let 120593 be a step-function of form 120593 = sum119899119896=11198871198961119905119896 Then we

extend ⟨sdot sdot⟩H119879 toT119879by defining

⟨119892 120593⟩H119879=

119899

sum

119896=1

119887119896int

119879

0

119892 (119904) 119877 (119905119896 d119904) (84)

In particular this implies that for 119892 and 120593 as above we have

⟨1198921119905 120593⟩

H119879= int

119905

0

119892 (119904) d ⟨1119904 120593⟩

H119879 (85)

We define extended domain Dom119864120575119879similarly as in [11]

Definition 46 A process 119906 isin 1198711(ΩT119879) belongs to Dom119864

120575119879

if10038161003816100381610038161003816E ⟨119906119863119865⟩H119879

10038161003816100381610038161003816le 119888

119906119865

2(86)

for any smooth random variable 119865 isin S In this case 120575(119906) isin1198712(Ω) is defined by duality relationship

E [119865120575 (119906)] = E ⟨119906119863119865⟩H119879 (87)

Remark 47 Note that in general Dom 120575119879and Dom119864

120575119879are

not comparable See [11] for discussion

Note now that if a function 119891 satisfies the growthcondition (61) then 1198911015840(119883

sdot)1

119905isin 119871

1(ΩT

119879) since (61) implies

E sup119905isin[0119879]

100381610038161003816100381610038161198911015840(119883

119905)10038161003816100381610038161003816

119901

lt infin (88)

for any 119901 lt (12120582)(sup119905isin[0119879]

119877(119905 119905))minus1 Consequently with

this definition we are able to get rid of the problem thatprocesses might not belong to corresponding H

119879-spaces

Furthermore this implies that the series expansion (69)converges in the norm 1198711(ΩT

119879) defined by

Eint119879

0

|119906 (119904)| |119877| (119905 d119904) (89)

which in turn implies (75) Hence it is straightforward toobtain the following by first showing the result for polynomi-als and then by approximating in a similar manner as done inSection 43 but using the extended domain instead

International Journal of Stochastic Analysis 11

Theorem 48 (Ito formula for extended Skorohod integrals)Let119883 be a separable centered Gaussian process with covariance119877 and assume that 119891 isin 1198622 satisfies growth condition (61)Furthermore assume that (H) holds and that the variance of119883is bounded and is of bounded variationThen for any 119905 isin [0 119879]the process 1198911015840(119883

sdot)1

119905belongs to Dom119864

120575119879and

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(90)

Remark 49 As an application of Theorem 48 it is straight-forward to derive version of Ito-Tanaka formula underadditional conditions which guarantee that for a certainsequence of functions 119891

119899we have the convergence of term

(12) int119905

011989110158401015840

119899(119883

119904)d119877(119904 119904) to the local time For details we refer

to [11] where authors derived such formula under theirassumptions

Finally let us note that the extension to functions 119891(119905 119909)is straightforward where 119891 satisfies the following growthcondition

max [1003816100381610038161003816119891 (119905 119909)1003816100381610038161003816 1003816100381610038161003816120597119905119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119909119891 (119905 119909)

1003816100381610038161003816]

le 119888119890120582|119909|2

(91)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1

Theorem 50 (Ito formula for extended Skorohod integralsII) Let 119883 be a separable centered Gaussian process withcovariance 119877 and assume that 119891 isin 11986212 satisfies growthcondition (91) Furthermore assume that (H) holds and thatthe variance of119883 is bounded and is of bounded variationThenfor any 119905 isin [0 119879] the process 120597

119909119891(sdot 119883

sdot)1

119905belongs to Dom119864

120575119879

and

119891 (119905 119883119905) = 119891 (0 119883

0) + int

119905

0

120597119909119891 (119904 119883

119904) 120575119883

119904

+ int

119905

0

120597119905119891 (119904 119883

119904) d119904

+1

2int

119905

0

120597119909119909119891 (119904 119883

119904) d119877 (119904 119904)

(92)

Proof Taking into account that we have no problems con-cerning processes to belong to the required spaces theformula follows by approximating with polynomials of form119901(119909)119902(119905) and following the proof of Theorem 40

5 Applications

We illustrate how some results transfer easily from theBrownian case to the Gaussian Fredholm processes

51 Equivalence in Law The transfer principle has alreadybeen used in connection with the equivalence of law of

Gaussian processes in for example [24] in the context offractional Brownian motions and in [2] in the context ofGaussianVolterra processes satisfying certain nondegeneracyconditions The following proposition uses the Fredholmrepresentation (8) to give a sufficient condition for theequivalence of general Gaussian processes in terms of theirFredholm kernels

Proposition 51 Let 119883 and be two Gaussian processes withFredholm kernels 119870

119879and

119879 respectively If there exists a

Volterra kernel ℓ isin 1198712([0 119879]2) such that

119879(119905 119904) = 119870

119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 (93)

then119883 and are equivalent in law

Proof Recall that by the Hitsuda representation theorem [5Theorem 63] a centered Gaussian process is equivalent inlaw to a Brownian motion on [0 119879] if and only if there existsa kernel ℓ isin 1198712([0 119879]2) and a Brownian motion119882 such that admits the representation

119905= 119882

119905minus int

119905

0

int

119904

0

ℓ (119904 119906) d119882119906d119904 (94)

Let119883 have the Fredholm representations

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (95)

Then is equivalent to119883 if it admits in law the representa-tion

119905

119889

= int

119879

0

119870119879(119905 119904) d

119904 (96)

where is connected to119882 of (95) by (94)In order to show (96) let

119905= int

119879

0

119879(119905 119904) d1198821015840

119905(97)

be the Fredholm representation of Here 1198821015840 is someBrownian motion Then by using connection (93) and theFubini theorem we obtain

119905= int

119879

0

119879(119905 119904) d1198821015840

119904

119889

= int

119879

0

119879(119905 119904) d119882

119904

= int

119879

0

(119870119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906) d119882

119904

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

10 International Journal of Stochastic Analysis

Proof Note that assumption is a Fredholm version of con-dition (K2) in [1] which implies condition (66) Hence theresult follows byTheorem 40

44 Extended Divergence Operator As shown in Section 43the Ito formula (60) is the only possibility However theproblem is that the space 1198712(ΩH

119879) may be too small to

contain the elements 1198911015840(119883sdot)1

119905 In particular it may happen

that not even the process 119883 itself belongs to 1198712(ΩH119879) (see

eg [7] for the case of fractional Brownian motion with119867 le 14) This problem can be overcome by consideringan extended domain of 120575

119879 The idea of extended domain is

to extend the inner product ⟨119906 120593⟩H119879 for simple 120593 to moregeneral processes 119906 and then define extended domain by(47) with a restricted class of test variables 119865 This also givesanother intuitive reason why extended domain of 120575

119879can be

useful indeed here we have proved that Ito formula (60) isthe only possibility and what one essentially needs for suchresult is the following

(i) 119883sdot1119905belongs to Dom 120575

119879

(ii) Equation (75) is valid for functions satisfying (61)

Consequently one should look for extensions of operator 120575119879

such that these two things are satisfiedTo facilitate the extension of domain we make the

following relatively moderate assumption

(H) The function 119905 997891rarr 119877(119905 119904) is of bounded variation on[0 119879] and

sup119905isin[0119879]

int

119879

0

|119877| (d119904 119905) lt infin (78)

Remark 44 Note that we are making the assumption on thecovariance 119877 not the kernel 119870

119879 Hence our case is different

from that of [1] Also [11] assumed absolute continuity in 119877we are satisfied with bounded variation

We will follow the idea from Lei and Nualart [11] andextend the inner product ⟨sdot sdot⟩H119879 beyondH

119879

Consider a step-function 120593Then on the one hand by theisometry property we have

⟨120593 1119905⟩H119879= int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 (79)

where 119892119905(119904) = 119870(119905 119904) isin 119871

2([0 119879]) On the other hand by

using adjoint property (see Remark 20) we obtain

int

119879

0

(119870lowast

119879120593) (119904) 119892

119905(119904) d119904 = int

119879

0

120593 (119904) (119870119879119892119905) (d119904) (80)

where computing formally we have

(119870119879119892119905) (d119904) = int

119879

0

119892119905(119906)119870

119879(d119904 119906) d119906

= int

119879

0

119870119879(119905 119906)119870

119879(d119904 119906) d119906 = 119877 (119905 d119904)

(81)

Consequently

⟨120593 1119905⟩H119879= int

119879

0

120593 (119904) 119877 (119905 d119904) (82)

This gives motivation to the following definition similar tothat of [11 Definition 21]

Definition 45 Denote by T119879the space of measurable func-

tions 119892 satisfying

sup119905isin[0119879]

int

119879

0

1003816100381610038161003816119892 (119904)1003816100381610038161003816 |119877| (119905 d119904) lt infin (83)

and let 120593 be a step-function of form 120593 = sum119899119896=11198871198961119905119896 Then we

extend ⟨sdot sdot⟩H119879 toT119879by defining

⟨119892 120593⟩H119879=

119899

sum

119896=1

119887119896int

119879

0

119892 (119904) 119877 (119905119896 d119904) (84)

In particular this implies that for 119892 and 120593 as above we have

⟨1198921119905 120593⟩

H119879= int

119905

0

119892 (119904) d ⟨1119904 120593⟩

H119879 (85)

We define extended domain Dom119864120575119879similarly as in [11]

Definition 46 A process 119906 isin 1198711(ΩT119879) belongs to Dom119864

120575119879

if10038161003816100381610038161003816E ⟨119906119863119865⟩H119879

10038161003816100381610038161003816le 119888

119906119865

2(86)

for any smooth random variable 119865 isin S In this case 120575(119906) isin1198712(Ω) is defined by duality relationship

E [119865120575 (119906)] = E ⟨119906119863119865⟩H119879 (87)

Remark 47 Note that in general Dom 120575119879and Dom119864

120575119879are

not comparable See [11] for discussion

Note now that if a function 119891 satisfies the growthcondition (61) then 1198911015840(119883

sdot)1

119905isin 119871

1(ΩT

119879) since (61) implies

E sup119905isin[0119879]

100381610038161003816100381610038161198911015840(119883

119905)10038161003816100381610038161003816

119901

lt infin (88)

for any 119901 lt (12120582)(sup119905isin[0119879]

119877(119905 119905))minus1 Consequently with

this definition we are able to get rid of the problem thatprocesses might not belong to corresponding H

119879-spaces

Furthermore this implies that the series expansion (69)converges in the norm 1198711(ΩT

119879) defined by

Eint119879

0

|119906 (119904)| |119877| (119905 d119904) (89)

which in turn implies (75) Hence it is straightforward toobtain the following by first showing the result for polynomi-als and then by approximating in a similar manner as done inSection 43 but using the extended domain instead

International Journal of Stochastic Analysis 11

Theorem 48 (Ito formula for extended Skorohod integrals)Let119883 be a separable centered Gaussian process with covariance119877 and assume that 119891 isin 1198622 satisfies growth condition (61)Furthermore assume that (H) holds and that the variance of119883is bounded and is of bounded variationThen for any 119905 isin [0 119879]the process 1198911015840(119883

sdot)1

119905belongs to Dom119864

120575119879and

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(90)

Remark 49 As an application of Theorem 48 it is straight-forward to derive version of Ito-Tanaka formula underadditional conditions which guarantee that for a certainsequence of functions 119891

119899we have the convergence of term

(12) int119905

011989110158401015840

119899(119883

119904)d119877(119904 119904) to the local time For details we refer

to [11] where authors derived such formula under theirassumptions

Finally let us note that the extension to functions 119891(119905 119909)is straightforward where 119891 satisfies the following growthcondition

max [1003816100381610038161003816119891 (119905 119909)1003816100381610038161003816 1003816100381610038161003816120597119905119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119909119891 (119905 119909)

1003816100381610038161003816]

le 119888119890120582|119909|2

(91)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1

Theorem 50 (Ito formula for extended Skorohod integralsII) Let 119883 be a separable centered Gaussian process withcovariance 119877 and assume that 119891 isin 11986212 satisfies growthcondition (91) Furthermore assume that (H) holds and thatthe variance of119883 is bounded and is of bounded variationThenfor any 119905 isin [0 119879] the process 120597

119909119891(sdot 119883

sdot)1

119905belongs to Dom119864

120575119879

and

119891 (119905 119883119905) = 119891 (0 119883

0) + int

119905

0

120597119909119891 (119904 119883

119904) 120575119883

119904

+ int

119905

0

120597119905119891 (119904 119883

119904) d119904

+1

2int

119905

0

120597119909119909119891 (119904 119883

119904) d119877 (119904 119904)

(92)

Proof Taking into account that we have no problems con-cerning processes to belong to the required spaces theformula follows by approximating with polynomials of form119901(119909)119902(119905) and following the proof of Theorem 40

5 Applications

We illustrate how some results transfer easily from theBrownian case to the Gaussian Fredholm processes

51 Equivalence in Law The transfer principle has alreadybeen used in connection with the equivalence of law of

Gaussian processes in for example [24] in the context offractional Brownian motions and in [2] in the context ofGaussianVolterra processes satisfying certain nondegeneracyconditions The following proposition uses the Fredholmrepresentation (8) to give a sufficient condition for theequivalence of general Gaussian processes in terms of theirFredholm kernels

Proposition 51 Let 119883 and be two Gaussian processes withFredholm kernels 119870

119879and

119879 respectively If there exists a

Volterra kernel ℓ isin 1198712([0 119879]2) such that

119879(119905 119904) = 119870

119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 (93)

then119883 and are equivalent in law

Proof Recall that by the Hitsuda representation theorem [5Theorem 63] a centered Gaussian process is equivalent inlaw to a Brownian motion on [0 119879] if and only if there existsa kernel ℓ isin 1198712([0 119879]2) and a Brownian motion119882 such that admits the representation

119905= 119882

119905minus int

119905

0

int

119904

0

ℓ (119904 119906) d119882119906d119904 (94)

Let119883 have the Fredholm representations

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (95)

Then is equivalent to119883 if it admits in law the representa-tion

119905

119889

= int

119879

0

119870119879(119905 119904) d

119904 (96)

where is connected to119882 of (95) by (94)In order to show (96) let

119905= int

119879

0

119879(119905 119904) d1198821015840

119905(97)

be the Fredholm representation of Here 1198821015840 is someBrownian motion Then by using connection (93) and theFubini theorem we obtain

119905= int

119879

0

119879(119905 119904) d1198821015840

119904

119889

= int

119879

0

119879(119905 119904) d119882

119904

= int

119879

0

(119870119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906) d119882

119904

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

International Journal of Stochastic Analysis 11

Theorem 48 (Ito formula for extended Skorohod integrals)Let119883 be a separable centered Gaussian process with covariance119877 and assume that 119891 isin 1198622 satisfies growth condition (61)Furthermore assume that (H) holds and that the variance of119883is bounded and is of bounded variationThen for any 119905 isin [0 119879]the process 1198911015840(119883

sdot)1

119905belongs to Dom119864

120575119879and

119891 (119883119905) = 119891 (119883

0) + int

119905

0

1198911015840(119883

119904) 120575119883

119904

+1

2int

119905

0

11989110158401015840(119883

119904) d119877 (119904 119904)

(90)

Remark 49 As an application of Theorem 48 it is straight-forward to derive version of Ito-Tanaka formula underadditional conditions which guarantee that for a certainsequence of functions 119891

119899we have the convergence of term

(12) int119905

011989110158401015840

119899(119883

119904)d119877(119904 119904) to the local time For details we refer

to [11] where authors derived such formula under theirassumptions

Finally let us note that the extension to functions 119891(119905 119909)is straightforward where 119891 satisfies the following growthcondition

max [1003816100381610038161003816119891 (119905 119909)1003816100381610038161003816 1003816100381610038161003816120597119905119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119891 (119905 119909)

1003816100381610038161003816 1003816100381610038161003816120597119909119909119891 (119905 119909)

1003816100381610038161003816]

le 119888119890120582|119909|2

(91)

for some 119888 gt 0 and 120582 lt (14)(sup0le119904le119879

E1198832

119904)minus1

Theorem 50 (Ito formula for extended Skorohod integralsII) Let 119883 be a separable centered Gaussian process withcovariance 119877 and assume that 119891 isin 11986212 satisfies growthcondition (91) Furthermore assume that (H) holds and thatthe variance of119883 is bounded and is of bounded variationThenfor any 119905 isin [0 119879] the process 120597

119909119891(sdot 119883

sdot)1

119905belongs to Dom119864

120575119879

and

119891 (119905 119883119905) = 119891 (0 119883

0) + int

119905

0

120597119909119891 (119904 119883

119904) 120575119883

119904

+ int

119905

0

120597119905119891 (119904 119883

119904) d119904

+1

2int

119905

0

120597119909119909119891 (119904 119883

119904) d119877 (119904 119904)

(92)

Proof Taking into account that we have no problems con-cerning processes to belong to the required spaces theformula follows by approximating with polynomials of form119901(119909)119902(119905) and following the proof of Theorem 40

5 Applications

We illustrate how some results transfer easily from theBrownian case to the Gaussian Fredholm processes

51 Equivalence in Law The transfer principle has alreadybeen used in connection with the equivalence of law of

Gaussian processes in for example [24] in the context offractional Brownian motions and in [2] in the context ofGaussianVolterra processes satisfying certain nondegeneracyconditions The following proposition uses the Fredholmrepresentation (8) to give a sufficient condition for theequivalence of general Gaussian processes in terms of theirFredholm kernels

Proposition 51 Let 119883 and be two Gaussian processes withFredholm kernels 119870

119879and

119879 respectively If there exists a

Volterra kernel ℓ isin 1198712([0 119879]2) such that

119879(119905 119904) = 119870

119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 (93)

then119883 and are equivalent in law

Proof Recall that by the Hitsuda representation theorem [5Theorem 63] a centered Gaussian process is equivalent inlaw to a Brownian motion on [0 119879] if and only if there existsa kernel ℓ isin 1198712([0 119879]2) and a Brownian motion119882 such that admits the representation

119905= 119882

119905minus int

119905

0

int

119904

0

ℓ (119904 119906) d119882119906d119904 (94)

Let119883 have the Fredholm representations

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (95)

Then is equivalent to119883 if it admits in law the representa-tion

119905

119889

= int

119879

0

119870119879(119905 119904) d

119904 (96)

where is connected to119882 of (95) by (94)In order to show (96) let

119905= int

119879

0

119879(119905 119904) d1198821015840

119905(97)

be the Fredholm representation of Here 1198821015840 is someBrownian motion Then by using connection (93) and theFubini theorem we obtain

119905= int

119879

0

119879(119905 119904) d1198821015840

119904

119889

= int

119879

0

119879(119905 119904) d119882

119904

= int

119879

0

(119870119879(119905 119904) minus int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906) d119882

119904

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

12 International Journal of Stochastic Analysis

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119879

119904

119870119879(119905 119906) ℓ (119906 119904) d119906 d119882

119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

int

119904

0

119870119879(119905 119904) ℓ (119904 119906) d119882

119906d119904

= int

119879

0

119870119879(119905 119904) d119882

119904

minus int

119879

0

119870119879(119905 119904) (int

119904

0

ℓ (119904 119906) d119882119906) d119904

= int

119879

0

119870119879(119905 119904) (d119882

119904minus int

119904

0

ℓ (119904 119906) d119882119906d119904)

= int

119879

0

119870119879(119905 119904) d

119904

(98)

Thus we have shown representation (96) and consequentlythe equivalence of and119883

52 Generalized Bridges We consider the conditioning orbridging of 119883 on 119873 linear functionals G

119879= [119866

119894

119879]119873

119894=1of its

paths

G119879(119883) = int

119879

0

g (119905) d119883119905= [int

119879

0

119892119894(119905) d119883

119905]

119873

119894=1

(99)

We assume without any loss of generality that the functions119892119894are linearly independent Also without loss of generality

we assume that 1198830= 0 and the conditioning is on the set

int119879

0g(119905)d119883

119905= 0 instead of the apparently more general

conditioning on the set int1198790g(119905)d119883

119905= y Indeed see in [16]

how to obtain the more general conditioning from this oneThe rigorous definition of a bridge is as follows

Definition 52 The generalized bridge measure Pg is theregular conditional law

Pg= P

g[119883 isin sdot] = P [119883 isin sdot | int

119879

0

g (119905) d119883119905= 0] (100)

A representation of the generalized Gaussian bridge is anyprocess119883g satisfying

P [119883gisin sdot] = P

g[119883 isin sdot]

= P [119883 isin sdot | int119879

0

g (119905) d119883119905= 0]

(101)

We refer to [16] for more details on generalized Gaussianbridges

There are many different representations for bridgesA very general representation is the so-called orthogonalrepresentation given by

119883g119905= 119883

119905minus ⟨⟨⟨1

119905 g⟩⟩⟩⊤ ⟨⟨⟨g⟩⟩⟩minus1 int

119879

0

g (119906) d119883119906 (102)

where by the transfer principle

⟨⟨⟨g⟩⟩⟩119894119895fl Cov [int

119879

0

119892119894(119905) d119883

119905 int

119879

0

119892119895(119905) d119883

119905]

= int

119879

0

119870lowast

119879119892119894(119905) 119870

lowast

119879119892119895(119905) d119905

(103)

A more interesting representation is the so-called canonicalrepresentation where the filtration of the bridge and theoriginal process coincide In [16] such representations wereconstructed for the so-called prediction-invertible Gaussianprocesses In this subsection we show how the transferprinciple can be used to construct a canonical-type bridgerepresentation for all Gaussian Fredholm processes We startwith an example that should make it clear how one uses thetransfer principle

Example 53 We construct a canonical-type representationfor 1198831 the bridge of 119883 conditioned on 119883

119879= 0 Assume

1198830= 0 Now by the Fredholm representation of 119883 we can

write the conditioning as

119883119879= int

119879

0

1 d119883119905= int

119879

0

119870119879(119879 119905) d119882

119905= 0 (104)

Let us then denote by 1198821lowast

the canonical representation ofthe Brownian bridge with conditioning (104) Then by [16Theorem 412]

d1198821lowast

119904= d119882

119904minus int

119904

0

119870119879(119879 119904) 119870

119879(119879 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904 (105)

Now by integrating against the kernel119870119879 we obtain from this

that

1198831

119905= 119883

119905

minus int

119879

0

119870119879(119905 119904) int

119904

0

119870119879(119879 119904) 119870

119879(T 119906)

int119879

119906119870119879(119879 V)2 dV

d119882119906d119904

(106)

This canonical-type bridge representation seems to be a newone

Let us then denote

⟨⟨g⟩⟩119894119895(119905) fl int

119879

119905

119892119894(119904) 119892

119895(119904) d119904 (107)

Then in the sameway as Example 53 by applying the transferprinciple to [16 Theorem 412] we obtain the followingcanonical-type bridge representation for general GaussianFredholm processes

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

International Journal of Stochastic Analysis 13

Proposition 54 Let 119883 be a Gaussian process with Fredholmkernel 119870

119879such that 119883

0= 0 Then the bridge 119883g admits the

canonical-type representation

119883g119905= 119883

119905minus int

119879

0

119870119879(119905 119904) int

119904

0

1003816100381610038161003816⟨⟨glowast⟩⟩1003816100381610038161003816 (119904) (g

lowast)⊤

(119904)

sdot ⟨⟨glowast⟩⟩minus1 (119904) glowast (119906)⟨⟨glowast⟩⟩ (119906)

d119882119906d119904

(108)

where glowast = 119870lowast

119879g

53 Series Expansions The Mercer square root (13) canbe used to build the Karhunen-Loeve expansion for theGaussian process 119883 But the Mercer form (13) is seldomknown However if one can find some kernel 119870

119879such that

representation (8) holds then one can construct a seriesexpansion for119883 by using the transfer principle ofTheorem 22as follows

Proposition 55 (series expansion) Let 119883 be a separableGaussian process with representation (8) Let (120601119879

119895)infin

119895=1be any

orthonormal basis on 1198712([0 119879]) Then 119883 admits the seriesexpansion

119883119905=

infin

sum

119895=1

int

119879

0

120601119879

119895(119904) 119870

119879(119905 119904) d119904 sdot 120585

119895 (109)

where (120585119895)infin

119895=1is a sequence of independent standard normal

random variables Series (109) converges in 1198712(Ω) and alsoalmost surely uniformly if and only if 119883 is continuous

The proof below uses reproducing kernel Hilbert spacetechnique For more details on this we refer to [25] wherethe series expansion is constructed for fractional Brownianmotion by using the transfer principle

Proof The Fredholm representation (8) implies immediatelythat the reproducing kernel Hilbert space of 119883 is the image1198701198791198712([0 119879]) and 119870

119879is actually an isometry from 1198712([0 119879])

to the reproducing kernel Hilbert space of 119883 The 1198712-expansion (109) follows from this due to [26 Theorem 37]and the equivalence of almost sure convergence of (109) andcontinuity of119883 follows [26 Theorem 38]

54 Stochastic Differential Equations and Maximum Likeli-hood Estimators Let us briefly discuss the following gener-alized Langevin equation

d119883120579

119905= minus120579119883

120579

119905d119905 + d119883

119905 119905 isin [0 119879] (110)

with some Gaussian noise 119883 parameter 120579 gt 0 and initialcondition119883

0 This can be written in the integral form

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119904 + int

119879

0

1119905(119904) d119883

119904 (111)

Here the integral int11987901119905(119904) d119883

119904can be understood in a

pathwise sense or in a Skorohod sense and both integrals

coincide Suppose now that the Gaussian noise 119883 has theFredholm representation

119883119905= int

119879

0

119870119879(119905 119904) d119882

119904 (112)

By applying the transfer principle we can write (111) as

119883120579

119905= 119883

120579

0minus 120579int

119905

0

119883120579

119904d119905 + int

119879

0

119870119879(119905 119904) d119882

119904 (113)

This equation can be interpreted as a stochastic differentialequation with some anticipating Gaussian perturbation termint119879

0119870119879(119905 119904)d119882

119904 Now the unique solution to (111) with an

initial condition119883120579

0= 0 is given by

119883120579

119905= 119890

minus120579119905int

119905

0

119890120579119904d119883

119904 (114)

By using integration by parts and by applying the Fredholmrepresentation of119883 this can be written as

119883120579

119905= int

119879

0

119870119879(119905 119906) d119882

119906

minus 120579int

119905

0

int

119879

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119882

119906d119904

(115)

which thanks to stochastic Fubinirsquos theorem can be writtenas

119883120579

119905= int

119879

0

[119870119879(119905 119906) minus 120579int

119905

0

119890minus120579119905119890120579119904119870119879(119904 119906) d119904] d119882

119906 (116)

In other words the solution 119883120579 is a Gaussian process with akernel

119870120579

119879(119905 119906) = 119870

119879(119905 119906) minus 120579int

119905

0

119890minus120579(119905minus119904)

119870119879(119904 119906) d119904 (117)

Note that this is just an example of how transfer principle canbe applied in order to study stochastic differential equationsIndeed for a more general equation

d119883119886

119905= 119886 (119905 119883

119886

119905) d119905 + d119883

119905(118)

the existence or uniqueness result transfers immediately tothe existence or uniqueness result of equation

119883119886

119905= 119883

119886

0+ int

119905

0

119886 (119904 119883119886

119904) d119904 + int

119879

0

119870119879(119905 119906) d119882

119906 (119)

and vice versaLet us end this section by discussing briefly how the

transfer principle can be used to build maximum likelihoodestimators (MLEs) for themean-reversal-parameter 120579 in (111)For details on parameter estimation in such equations withgeneral stationary-increment Gaussian noise we refer to [27]and references therein Let us assume that the noise119883 in (111)is infinite-generate in the sense that the Brownianmotion119882in its Fredholm representation is a linear transformation of

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 14: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

14 International Journal of Stochastic Analysis

119883 Assume further that the transformation admits a kernelso that we can write

119882119905= int

119879

0

119870minus1

119879(119905 119904) d119883

119904 (120)

Then by operating with the kernels 119870119879and 119870minus1

119879 we see that

(111) is equivalent to the anticipating equation

119882120579

119905= 119860

119879119905(119882

120579) +119882

119905 (121)

where

119860119879119905(119882

120579) = minus120579∬

119879

0

119870minus1

119879(119905 119904) 119870

119879(119904 119906) d119882120579

119906d119904 (122)

Consequently the MLE for (111) is the MLE for (121) whichin turn can be constructed by using a suitable anticipatingGirsanov theoremThere is a vast literature on how to do thissee for example [28] and references therein

6 Conclusions

Wehave shown that virtually everyGaussian process admits aFredholm representation with respect to a Brownian motionThis apparently simple fact has as far as we know remainedunnoticed until now The Fredholm representation immedi-ately yields the transfer principle which allows one to transferthe stochastic analysis of virtually any Gaussian process intostochastic analysis of the Brownian motion We have shownhow this can be done Finally we have illustrated the powerof the Fredholm representation and the associated transferprinciple in many applications

Stochastic analysis becomes easy with the Fredholmrepresentation The only obvious problem is to construct theFredholm kernel from the covariance function In principlethis can be done algorithmically but analytically it is verydifficultThe opposite construction is however trivialThere-fore if one begins the modeling with the Fredholm kerneland not with the covariance onersquos analysis will be simpler andmuch more convenient

Competing Interests

The authors declare that there is no conflict of interestregarding the publication of this paper

Acknowledgments

Lauri Viitasaari was partially funded by Emil Aaltonen Foun-dation Tommi Sottinen was partially funded by the FinnishCultural Foundation (National Foundationsrsquo Professor Pool)

References

[1] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[2] F Baudoin and D Nualart ldquoEquivalence of Volterra processesrdquoStochastic Processes and Their Applications vol 107 no 2 pp327ndash350 2003

[3] F Biagini Y Hu B Oslashksendal and T Zhang Stochastic Calculusfor Fractional BrownianMotion and Applications Springer NewYork NY USA 2008

[4] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[5] T Hida and M Hitsuda Gaussian Processes vol 120 of Trans-lations of Mathematical Monographs American MathematicalSociety Providence RI USA 1993

[6] E Azmoodeh T Sottinen L Viitasaari and A Yazigi ldquoNeces-sary and sufficient conditions for Holder continuity of Gaussianprocessesrdquo Statistics amp Probability Letters vol 94 pp 230ndash2352014

[7] P Cheridito and D Nualart ldquoStochastic integral of divergencetype with respect to the fractional Brownian motion with Hurstparameter 119867120598(0 12)rdquo Annales de lrsquoInstitut Henri Poincare (B)Probability and Statistics vol 41 no 6 pp 1049ndash1081 2005

[8] I Kruk F Russo and C A Tudor ldquoWiener integrals Malliavincalculus and covariance measure structurerdquo Journal of Func-tional Analysis vol 249 no 1 pp 92ndash142 2007

[9] I Kruk and F Russo ldquoMalliavin-Skorohod calculus andPaley-Wiener integral for covariance singular processesrdquohttparxivorgabs10116478

[10] O Mocioalca and F Viens ldquoSkorohod integration and stochas-tic calculus beyond the fractional Brownian scalerdquo Journal ofFunctional Analysis vol 222 no 2 pp 385ndash434 2005

[11] P Lei and D Nualart ldquoStochastic calculus for Gaussian pro-cesses and applications to hitting timesrdquo Communications onStochastic Analysis vol 6 no 3 pp 379ndash402 2012

[12] J Lebovits ldquoStochastic calculus with respect to Gaussian pro-cessesrdquo 2014 httparxivorgabs14081020

[13] V Pipiras and M S Taqqu ldquoAre classes of deterministicintegrands for fractional Brownian motion on an intervalcompleterdquo Bernoulli vol 7 no 6 pp 873ndash897 2001

[14] C Bender and R J Elliott ldquoOn the Clark-Ocone theorem forfractional Brownian motions with Hurst parameter bigger thana halfrdquo Stochastics and Stochastics Reports vol 75 no 6 pp 391ndash405 2003

[15] H Cramer ldquoOn the structure of purely non-deterministicstochastic processesrdquo Arkiv for Matematik vol 4 pp 249ndash2661961

[16] T Sottinen and A Yazigi ldquoGeneralized Gaussian bridgesrdquoStochastic Processes and Their Applications vol 124 no 9 pp3084ndash3105 2014

[17] K Ito ldquoMultiple Wiener integralrdquo Journal of the MathematicalSociety of Japan vol 3 pp 157ndash169 1951

[18] S T Huang and S Cambanis ldquoStochastic and multiple Wienerintegrals for Gaussian processesrdquoThe Annals of Probability vol6 no 4 pp 585ndash614 1978

[19] A Dasgupta and G Kallianpur ldquoMultiple fractional integralsrdquoProbabilityTheory andRelated Fields vol 115 no 4 pp 505ndash5251999

[20] ADasgupta andGKallianpur ldquoChaos decomposition ofmulti-ple fractional integrals and applicationsrdquo ProbabilityTheory andRelated Fields vol 115 no 4 pp 527ndash548 1999

[21] V Perez-Abreu and C Tudor ldquoMultiple stochastic fractionalintegrals a transfer principle for multiple stochastic fractionalintegralsrdquo Boletın de la Sociedad Matematica Mexicana vol 8no 2 pp 187ndash203 2002

[22] D NualartTheMalliavin Calculus and Related Topics Probabil-ity and Its Applications Springer New York NY USA 2006

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 15: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

International Journal of Stochastic Analysis 15

[23] I Nourdin and G Peccati ldquoSteinrsquos method and exact Berry-Esseen asymptotics for functionals of Gaussian fieldsrdquo TheAnnals of Probability vol 37 no 6 pp 2231ndash2261 2009

[24] T Sottinen ldquoOn Gaussian processes equivalent in law tofractional Brownian motionrdquo Journal of Theoretical Probabilityvol 17 no 2 pp 309ndash325 2004

[25] H Gilsing and T Sottinen ldquoPower series expansions forfractional Brownian motionsrdquo Theory of Stochastic Processesvol 9 no 3-4 pp 38ndash49 2003

[26] R Adler An Introduction to Continuity Extrema and RelatedTopics forGeneral Gaussian Processes vol 12 of Institute ofMath-ematical Statistics Lecture NotesmdashMonograph Series Institute ofMathematical Statistics Hayward Calif USA 1990

[27] T Sottinen and L Viitasaari ldquoParameter estimation for theLangevin equation with stationary-increment Gaussian noiserdquohttpsarxivorgabs160300390

[28] J P N Bishwal ldquoMaximum likelihood estimation in Skorohodstochastic differential equationsrdquo Proceedings of the AmericanMathematical Society vol 138 no 4 pp 1471ndash1478 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 16: Research Article Stochastic Analysis of Gaussian Processes ...downloads.hindawi.com/archive/2016/8694365.pdfkernel . Kruk et al. [ ] developed stochastic calculus for processeshavingn

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The KRUK Group

The KRUK Group

Sport sparring in Taekwon-Do ITF by Dr Zibby Kruk

Sport sparring in Taekwon-Do ITF by Dr Zibby Kruk

Bruce Kruk - Wet Fly Swing

Bruce Kruk - Wet Fly Swing

Kruk - Of Fat People and Fundamental Rights

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KRUK results presentation 1H2017 - Customers income (PLN -36.2m)in balancesheet, instead of P&L (PLN +35.4m) As a result, KRUK restated its Q1 2017 results. Comment PLNm Q1 2017 (original

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Quality Control of 19th Century Weather Data · Michael Kruk, Raymond Truesdell, and Diana O’Connell January 2011 ...

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Presentation of 2018 full-year results KRUK Group · 2019-03-08 · Increase in net profit In 2018, the KRUK Group reported net profit of PLN 330m, which represents an increase of

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Introduction to UNIX/Linuxfolding.chemistry.msstate.edu/files/bootcamp/2018/session-03.pdfkernel; readily supports Linux-like shell scripting and windowing environment (true POSIX)

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Presentation of H1 2019 results KRUK Groupsecured part of the debt portfolio, KRUK recognised a negative revaluation of PLN 11,7m. • As a result of two partially offsetting effects,

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17.237 Developmental Psychology from Adolescence to Old Age Richard Kruk.

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PRESENTATION OF RESULTS - KRUK · 2016. 10. 18. · In Q1-Q3 2012, the KRUK Group purchased 36 debt portfolios, including 23 retail debt portfolios from financial institutions, 5

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Stochastic Optimization Online and batch stochastic ...ibis.t.u-tokyo.ac.jp/.../nagoya_intensive/Nagoya2015_3_StochasticOpt… · 1 Online stochastic optimization Stochastic gradient

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