CONVERGENCE TO WEIGHTED FRACTIONAL … TO WEIGHTED FRACTIONAL BROWNIAN SHEETS* JOHANNA GARZON ......

CONVERGENCE TO WEIGHTED FRACTIONALBROWNIAN SHEETS*

JOHANNA GARZON

Abstract. We define weighted fractional Brownian sheets, which are a classof Gaussian random fields with four parameters that include fractional Brown-ian sheets as special cases, and we give some of their properties. We show thatfor certain values of the parameters the weighted fractional Brownian sheetsare obtained as limits in law of occupation time fluctuations of a stochasticparticle model. In contrast with some known approximations of fractionalBrownian sheets which use a kernel in a Volterra type integral representationof fractional Brownian motion with respect to ordinary Brownian motion, ourapproximation does not make use of a kernel.

1. Introduction

Fractional Brownian sheets have been studied by several authors for their math-ematical interest and their applications. One of the first papers on the subject is[12]. Some types of approximations of fractional Brownian sheets have been ob-tained recently (e.g., [2], [3], [9], [13], [14], [15]). In this paper we give a new typeof approximation for certain values of the parameters by means of occupation timefluctuations of a stochastic particle model. The limits that are obtained in thisway are a more general class of Gaussian random fields.

We consider centered Gaussian random fields W = (Ws,t)s,t≥0 with parameters(ai, bi), i = 1, 2, whose covariance is given by

KW ((s, t), (s′, t′)) = E (Ws,tWs′,t′) = C(1)(s, s′)C(2)(t, t′), (1.1)

where each C(i) is of the form

C(i)(u, v) =∫ u∧v

0

rai [(u− r)bi + (v − r)bi ]dr, i = 1, 2, (1.2)

with the following ranges for the parameters:

ai > −1, −1 < bi ≤ 1, |bi| ≤ 1 + ai. (1.3)

C(i) is the covariance of weighted fractional Brownian motion with parameters(ai, bi). Weighted fractional Brownian motions were introduced in [7]. We call Wa weighted fractional Brownian sheet with parameters (ai, bi), i = 1, 2. In the casea1 = a2 = 0 (the weight functions are 1) W is a fractional Brownian sheet with

2000 Mathematics Subject Classification. Primary 60G60; Secondary 60G15, 60F05.Key words and phrases. Fractional Brownian sheet, weighted fractional Brownian sheet,

approximation in law, long-range dependence.* Research partially supported by CONACYT grant 45684-F.

1

Serials Publications www.serialspublications.com

Communications on Stochastic Analysis Vol. 3, No. 1 (2009) 1-14

2 JOHANNA GARZON

parameters ( 12 (1 + b1), 1

2 (1 + b2)). The case a1 = a2 = b1 = b2 = 0 corresponds tothe ordinary Brownian sheet. If b1 = b2 = 0, and at least one ai is not 0, then Wis a time-inhomogeneous Brownian sheet.

Due to the covariance structure (1.1), (1.2), many properties of W are con-sequences of those of weighted fractional Brownian motion. We will prove anapproximation in law of W for ai and bi of the form ai = −γi/αi, bi = 1 − 1/αi,with 0 ≤ γi < 1 and 1 < αi ≤ 2; hence the approximation is restricted to valuesof ai and bi such that −1 < ai ≤ 0 and 0 < bi < 1 + ai, i = 1, 2.

The approximations of fractional Brownian sheets in [2], [15] are based on aPoisson random measure on R+ × R+ and a kernel which appears in a Volterratype integral representation of fractional Brownian motion with respect to ordinaryBrownian motion. The approximation in [3], analogous to the functional invariancetheorem, also uses the kernel. Our approach does not use a kernel. We also usea Poisson random measure, but on R × R instead of R+ × R+ and in a differentway from [2], [15]. Some of the other approximations cited above are motived bysimulation of fractional Brownian sheets. Our approximation is not intended forsimulation, but rather to show that weighted fractional Brownian sheets emergein a natural way from a simple particle model.

In section 2 we give the properties of W , in particular long-range dependence.In section 3 we describe the particle system and we prove convergence to W ofrescaled ocupation time fluctuations of the system for the above mentioned valuesof the parameters.

2. Properties

We consider R2+ with the following partial order: for z = (s, t) and z′ = (s′, t′),

z ¹ z′ iff s ≤ s′ and t ≤ t′, z ≺ z′ iff s < s′ and t < t′, and if z ≺ z′ we denoteby (z, z′] the rectangle (s, s′] × (t, t′]. We refer to elements of R2

+ as “times” forsimplicity of exposition.

If X = (Xz)z∈R2+

is a two-time stochastic process, the increment of X over therectangle (z, z′] with z = (s, t), z′ = (s′, t′) is defined by

X((z, z′]) ≡ ∆s,tX(s′, t′) := X(s′,t′) −X(s,t′) −X(s′,t) +X(s,t).

We denote the covariance of the increments of the process X over the rectangles((s, t), (s′, t′)], ((p, r), (p′, r′)] by

KX ((s, t), (s′, t′); (p, r), (p′, r′)) = Cov (∆s,tX(s′, t′),∆p,rX(p′, r′)) .

The covariance of W over rectangles is given by

KW ((s, t), (s′, t′); (p, r), (p′, r′))

= (C(1)(s′, p′)− C(1)(s, p′)− C(1)(s′, p) + C(1)(s, p))

× (C(2)(t′, r′)− C(2)(t, r′)− C(2)(t′, r) + C(2)(t, r))

= Cov(Y (1)s′ − Y (1)

s , Y(1)p′ − Y (1)

p )Cov(Y (2)t′ − Y

(2)t , Y

(2)r′ − Y (2)

r ), (2.1)

where Y (i) is weighted fractional Brownian motion with parameters (ai, bi), i =1, 2.

CONVERGENCE TO WEIGHTED FRACTIONAL BROWNIAN SHEETS 3

The next theorem contains some properties of weighted fractional Browniansheets.

Theorem 2.1. The weighted fractional Brownian sheet W with parameters (ai, bi),i = 1, 2, has the following properties:

(1) Self-similarity:

(Whs,kt)s,t≥0

d= h(1+a1+b1)/2k(1+a2+b2)/2(Ws,t)s,t≥0 for each h, k > 0, (2.2)

where d= denotes equality in distribution.(2) W has stationary increments only in the case a1 = a2 = 0.(3) Covariance of increments: For (0, 0) ¹ (s, t) ≺ (s′, t′) ¹ (p, r) ≺ (p′, r′),

KW ((s, t), (s′, t′); (p, r), (p′, r′))

=∫ s′

s

ua1 [(p′ − u)b1 + (p− u)b1 ]du∫ t′

t

va2 [(r′ − v)b2 + (r − v)b2 ]dv, (2.3)

hence

KW ((s, t), (s′, t′); (p, r), (p′, r′))

> 0 if b1b2 > 0,= 0 if b1b2 = 0,< 0 if b1b2 < 0.

(4) The one-time processes (Ws,t)s≥0 (t fixed) and (Ws,t)t≥0 (s fixed) areweighted fractional Brownian motions (multiplied by constants) with pa-rameters (a1, b1) and (a2, b2), respectively.

(5)

E((∆s,tWs′,t′)2) = 4

∫ s′

s

ua1(s′ − u)b1du

∫ t′

t

va2(t′ − v)b2dv. (2.4)

(6)

limε,δ→0

ε−b1−1δ−b2−1E((∆s,tWs+ε,t+δ)2) =

4(1 + b1)(1 + b2)

sa1ta2 , (2.5)

limT,S→∞

S−(1+a1+b1)T−(1+a2+b2)E((∆s,tWs+S,t+T )2)

= 4∫ 1

0

ua1(1− u)b1du

∫ 1

0

va2(1− v)b2dv, (2.6)

hence W has asymptotically stationary increments for long increments inR2

+, but not for short ones (if a1, a2 6= 0).(7) The finite-dimensional distributions of the process

(S−a1/2T−a2/2∆S,TWs+S,t+T )s,t≥0

converge as T, S →∞ to those of fractional Brownian sheet with parame-ters ( 1

2 (1 + b1), 12 (1 + b2)) multiplied by 2/[(1 + b1)(1 + b2)]1/2.

4 JOHANNA GARZON

(8) Long-range dependence: for (s, t) ≺ (s′, t′), (p, u) ≺ (p′, u′)

limτ,κ→∞

τ1−b1κ1−b2KW ((s, t), (s′, t′); (p+ τ, u+ κ), (p′ + τ, u′ + κ))

=b1b2

(1 + a1)(1 + a2)(p′ − p)((s′)1+a1 − s1+a1)(u′ − u)((t′)1+a2 − t1+a2). (2.7)

(9) For θ > 0, we define the one-time process (Zt)t≥0 = (Wt,θt)t≥0, i.e., thesheet restricted to a ray through the origen. (Note that Z is not a weightedfractional Brownian motion.) Then for 0 ≤ bi < 1, i = 1, 2, and not bothb1, b2 equal to 0, this process has the long-range dependence property

limτ→∞

τ1−(b1+b2)Cov(Zv − Zu, Zt+τ − Zs+τ )

=θ1+a2+b2(b1 + b2)(1 + a1)(1 + a2)

(v2+a1+a2 − u2+a1+a2)(t− s), u < v, s < t. (2.8)

Proof. Except for part (9), the proofs follow directly from the form of KW givenby (1.1), (1.2) and properties of weighted fractional Brownian motion [7]. We givean outline of the proof of part (9).

We have, for u < v, s < t,

Cov(Zv − Zu, Zt+τ − Zs+τ )

= θ1+a2+b2 [C(1)(v, t+ τ)C(2)(v, t+ τ)− C(1)(v, s+ τ)C(2)(v, s+ τ)

− C(1)(u, t+ τ)C(2)(u, t+ τ) + C(2)(u, s+ τ)C(2)(u, s+ τ)], (2.9)

C(1)(v, t+ τ)C(2)(v, t+ τ)− C(1)(v, s+ τ)C(2)(v, s+ τ)

= [C(1)(v, t+ τ)− C(1)(v, s+ τ)]C(2)(v, t+ τ)

+ [C(2)(v, t+ τ)− C(2)(v, s+ τ)]C(1)(v, s+ τ)

=∫ v

0

ra1 [(t− r + τ)b1 − (s− r + τ)b1 ]dr∫ v

0

ra2 [(t− r + τ)b2 + (v − r)b2 ]dr

+∫ v

0

ra2 [(t− r + τ)b2 − (s− r + τ)b2 ]dr∫ v

0

ra1 [(s− r + τ)b1 + (v − r)b1 ]dr,

(2.10)

and similarly for the last two terms. The result follows from (2.9), (2.10) and thelimits

limτ→∞

τ1−b[(t2 + τ)b − (t1 + τ)b] = b(t2 − t1)

and

limτ→∞

τ−b

∫ v

0

ra[(t+ τ)b + (v − r)b]dr =v1+a

1 + a.

¤

Remark 2.2. There are three different long-range dependence regimes in property(9), and they are independent of a1, a2. The covariance of increments of Z has apower decay for b1 + b2 < 1, a power growth for b1 + b2 > 1, and a non-triviallimit for b1 + b2 = 1. We do not know if this property has been noted beforefor fractional Brownian sheets. It is worthwhile to observe that the non-Gaussian


process (Y (1)t Y

(2)θt )t≥0, where Y (i) are independent weighted fractional Brownian

motions with parameters (ai, bi), i = 1, 2, has the same long-range dependencebehavior.

In [7] it is shown that As,t =∫ t

sua(t − u)bdu, 0 ≤ s < t, has the following

bounds: If a ≥ 0, s, t ≤ T for any T > 0 and constant M = M(T ), and also ifa < 0, s, t ≥ ε for any ε > 0 and constant M = M(ε),

As,t ≤M |t− s|1+b.

If a < 0, 1 + a+ b > 0, s, t ≥ 0,

As,t ≤M |t− s|1+a+b.

Then it follows from (2.4) that for 0 < ε ≤ s < s′ < T, 0 < ε ≤ t < t′ < T andi = 1, 2,

E((∆s,tWs′,t′)2) ≤M (s′ − s)δ1 (t′ − t)δ2 , (2.11)

where

δi =

1 + ai + bi if ai < 0 and 1 + ai + bi > 0,1 + bi otherwise. (2.12)

The next lemma allows us to prove the continuity of W .

Lemma 2.3. [1], [10] Let X = (Xs,t)s,t≥0 be a two-time stochastic process on aprobability space (Ω,F, P ) which is null almost surely on the axes and such thatthere exist p > 0, a, b ∈ (1/p,∞), such that

(E(|∆s,tXs+h,t+k|p))1/p ≤M |h|a|k|b.Then X has a modification X with continuous trajectories. Also, the trajectoriesof X are Holder with exponents (a′, b′), for a′ ∈ (0, a− 1/p), b′ ∈ (0, b− 1/p), thatis, for any ω ∈ Ω exists Mω > 0 such that for any s, s′, t, t′,

|∆s,tXs′,t′(ω)| ≤Mω(s′ − s)a′(t′ − t)b′ , s < s′, t < t′.

Proposition 2.4. The weighted fractional Brownian sheet (Ws,t)s,t≥0 has a mod-ification (Ws,t)s,t≥0 with continuous trajectories. Also, the trajectories of W areHolder with exponents (x, y) for any x ∈ (0, 1

2δ1), y ∈ (0, 12δ2), where δi are as in

(2.12).

Proof. From the moments of the normal distribution and equations (2.4) and (2.11)we have

(E(|∆s,tWs+h,t+k|r))1/r

= C

(∫ s+h

s

ua1(s+ h− u)b1du

∫ t+k

t

va2(t+ k − v)b2dv

)1/2

≤Mhδ1/2kδ2/2,

with some constants C and M . Taking r > max 2/δ1, 2/δ2 we have the condi-tions of Lemma 2.3, and the result follows. ¤

6 JOHANNA GARZON

3. Approximation

The random field W , for some values of the parameters ai, bi, arises as a limitin distribution of occupation time fluctuations of a system of particles of two typesthat move as pairs in R×R according to independent stable Levy processes. Thesystem is described as follows. Given a Poisson random measure on R × R withintensity measure µ, N0,0 = Pois(µ), from each point (x1, x2) of N0,0 come outtwo independent Levy processes, from x1 comes out ξx1 , symmetric α1-stable, andfrom x2 comes out ζx2 , symmetric α2-stable (0 < αi ≤ 2, i = 1, 2). Let N =(Nu,v)u,v≥0 denote random measure process on R × R such that Nu,v representsthe configuration of particles at time (u, v),

Nu,v =∑

(x1,x2)∈N0,0

δ(ξx1u ,ζ

x2v ) =

∑

(x1,x2)∈N0,0

δξx1u⊗ δζx2

v. (3.1)

For ϕ,ψ ∈ L1(R) (ϕ,ψ 6= 0) fixed, we write

〈Nu,v, ϕ⊗ ψ〉 =∑

(x1,x2)∈N0,0

〈δξx1u⊗ δζx2

v, ϕ⊗ ψ〉 =

∑

(x1,x2)∈N0,0

ϕ(ξx1u )ψ(ζx2

v ). (3.2)

We define the occupation time process of N by

〈Ls,t, ϕ⊗ ψ〉 =∫ s

0

∫ t

0

〈Nu,v, ϕ⊗ ψ〉 dvdu, s, t ≥ 0, (3.3)

and the rescaled occupation time fluctuation process by

XT (s, t) =1FT

(〈LTs,T t, ϕ⊗ ψ〉 − E(〈LTs,T t, ϕ⊗ ψ〉)), s, t ≥ 0, (3.4)

where T is the time scaling and FT is a norming. We choose the intensity measureµ for the Poisson initial particle configuration as

µ(dx1, dx2) = µ1 ⊗ µ2(dx1, dx2) = µ1(dx1)µ2(dx2),

withµi(dxi) = dxi/ |xi|γi , 0 ≤ γi < 1, i = 1, 2. (3.5)

The homogeneous case corresponds to γ1 = γ2 = 0 and it gives rise to the usualfractional Brownian sheet. We will show that for

FT = F(1)T F

(2)T with F (i)

T = T 1−(1+γi)/2αi , 0 ≤ γi < 1 < αi, i = 1, 2, (3.6)

the finite-dimensional distributions of the process XT converge in law as T → ∞to those of weighted fractional Brownian sheet with parameters ai = −γi/αi,bi = 1− 1/αi, i = 1, 2. In the case a1 = a2 = 0 we will also prove tightness.

Theorem 3.1. If XT is the process defined in (3.4), 0 ≤ γi < 1 < αi, i = 1, 2,with FT defined by (3.6), then the finite-dimensional distributions of XT convergeas T → ∞ to the finite-dimensional distributions of DW , where W is weightedfractional Brownian sheet with parameters a1 = −γ1/α1, b1 = 1 − 1/α1, a2 =−γ2/α2, b2 = 1− 1/α2, and D is the constant

D =∫

Rϕ(x)dx

∫

Rψ(x)dx

( 2∏

i=1

11− 1/αi

pαi1 (0)

(∫

R

pαi1 (x)|x|γi

dx

))1/2

, (3.7)


where pαt (x) is the density of the symmetric α-stable Levy process, which is given

by

pαt (x) =

12π

∫

Rexp − (ixy + t|y|α) dy.

Proof. For each k ∈ N, d1, · · · , dk ∈ R and (s1, t1), · · · , (sk, tk) ∈ R2+, we must

show thatk∑

j=1

djXTsj ,tj

converges in law tok∑

j=1

djWsj ,tjas T →∞,

which we do by proving convergence of the corresponding characteristic functions.From the fact that N0,0 = Pois(µ1 ⊗ µ2), we have for each θ ∈ R,

CT (θ) := E expiθ

k∑

j=1

djXTsj ,tj

= exp− iθ

FT

k∑

j=1

djE(〈LTsj ,tj

, ϕ⊗ ψ〉)

× exp−

∫

R×R

[1−E(x1,x2)

(exp

iθ

FT

k∑

j=1

dj〈LTsj ,tj

, ϕ⊗ ψ〉)]

µ1(dx1)µ2(dx2),

(3.8)

where E(x1,x2) denotes expectation starting with one pair of initial particles in(x1, x2), (see e.g., [11], mixed Poisson process).

We also need the mean and the covariance of N . From the Poisson initialcondition, the first and second moments are given by

E (〈Nu,v, ϕ⊗ ψ〉) =∫

R×RE(x1,x2) (〈Nu,v, ϕ⊗ ψ〉)µ1(dx1)µ2(dx2)

=∫

R×RE (ϕ(ξx1

u )ψ(ζx2v ))µ1(dx1)µ2(dx2) (3.9)

and

E (〈Nu1,v1 , ϕ⊗ ψ〉〈Nu2,v2 , ϕ⊗ ψ〉)

=∫

R×RE(x1,x2) (〈Nu1,v1 , ϕ⊗ ψ〉〈Nu2,v2 , ϕ⊗ ψ〉)µ1(dx1)µ2(dx2)

+∫

R×RE(x1,x2) (〈Nu1,v1 , ϕ⊗ ψ〉)µ1(dx1)µ2(dx2)

×∫

R×RE(x1,x2) (〈Nu2,v2 , ϕ⊗ ψ〉)µ1(dx1)µ2(dx2)

=∫

R×RE

(ϕ(ξx1

u1)ψ(ζx2

v1)ϕ(ξx1

u2)ψ(ζx2

v2))µ1(dx1)µ2(dx2)

+∫

R×RE

(ϕ(ξx1

u1)ψ(ζx2

v1))µ1(dx1)µ2(dx2)

∫

R×RE

(ϕ(ξx1

u2)ψ(ζx2

v2))µ1(dx1)µ2(dx2),

8 JOHANNA GARZON

hence, by the independence of ξ and ζ, and the Markov property,

Cov (〈Nu1,v1 , ϕ⊗ ψ〉 , 〈Nu2,v2 , ϕ⊗ ψ〉)

=∫

R×RE(x1,x2) (〈Nu1,v1 , ϕ⊗ ψ〉〈Nu2,v2 , ϕ⊗ ψ〉)µ1(dx1)µ2(dx2)

=∫

RT α1

u1∧u2(ϕT α1

|u1−u2|ϕ)(x1)µ1(dx1)∫

RT α2

v1∧v2(ψT α2

|v1−v2|ψ)(x2)µ2(dx2), (3.10)

where T αit denotes the semigroup of the symmetric αi-stable process.

Using an expansion of the characteristic function (see e.g., [5], p. 297) in theintegrand with respect to (x1, x2) in (3.8), it is equal to

1 +iθ

FTE(x1,x2)

( k∑

j=1

dj〈LTsj ,tj

, ϕ⊗ ψ〉)− θ2

2F 2T

E(x1,x2)

( k∑

j=1

dj〈LTsj ,tj

, ϕ⊗ ψ〉)2

+ δT(x1,x2)

,

where

|δT(x1,x2)

| ≤ θ3

F 3T

E(x1,x2)

( k∑

j=1

dj〈LTsj ,tj

, ϕ⊗ ψ〉)3

. (3.11)

Sincek∑

j=1

djE(〈LTsj ,tj

, ϕ⊗ ψ〉) =∫

R×R

k∑

j=1

djE(x1,x2)(〈LTsj ,tj

, ϕ⊗ ψ〉)µ1(dx1)µ2(dx2),

then (3.8) becomes

CT (θ) = exp−

∫

R×R

[θ2

2F 2T

E(x1,x2)

( k∑

j=1

dj〈LTsj ,tj

, ϕ⊗ ψ〉)2

+ δT(x1,x2)

]µ1(dx1)µ2(dx2)

(3.12)

and by (3.3) and a previous calculation,∫

R×R

1F 2

T

E(x1,x2)

( k∑

j=1

dj〈LTs,t, ϕ⊗ ψ〉

)2

µ1(dx1)µ2(dx2)

=1F 2

T

k∑

j=1

dj

k∑

j′=1

dj′

∫

R×R

∫ Tsj

0

∫ Ttj

0

∫ Tsj′

0

∫ Ttj′

0

E(x1,x2) (〈Nu1,v1 , ϕ⊗ ψ〉〈Nu2,v2 , ϕ⊗ ψ〉) dv2du2dv1du1µ1(dx1)µ2(dx2)

=k∑

j=1

dj

k∑

j′=1

dj′1

F(1)2T

∫

R

∫ Tsj

0

∫ Tsj′

0

T α1u1∧u2

(ϕT α1|u1−u2|ϕ)(x1)du2du1

dx1

|x1|γ1

× 1

F(2)2T

∫

R

∫ Ttj

0

∫ Ttj′

0

T α2v1∧v2

(ψT α2|v1−v2|ψ)(x2)dv2dv1

dx2

|x2|γ2. (3.13)


Now, recalling (3.6) we have

1(T 1−(1+γ)/2α

)2

∫

R

∫ Ts1

0

∫ Ts2

0

T αu1∧u2

(ϕT α|u1−u2|ϕ)(x)du2du1

dx

|x|γ

=1(

T 1−(1+γ)/2α)2

∫

R

∫ Ts1

0

∫ Ts2

0

∫

Rpα

u1∧u2(x− y)ϕ(y)

×∫

Rpα|u1−u2|(y − z)ϕ(z)dzdydu2du1

dx

|x|γ ,

substituting u1 = Tu′1, u2 = Tu′2, using the self-similarity of the α-stable processin R, i.e., pα

t (x) = t−1/αpα1 (t−1/αx), and then substituting x = (T (u′1 ∧ u′2))1/α

x′,

= T (1+γ)/α

∫

R

∫ s1

0

∫ s2

0

∫

Rpα

T (u′1∧u′2)(x− y)ϕ(y)

×∫

Rpα

T |u′1−u′2|(y − z)ϕ(z)dzdydu′2du′1

dx

|x|γ

= T (γ−1)/α

∫

R

∫ s1

0

∫ s2

0

(u′1 ∧ u′2)−1/α|u′1 − u′2|−1/α

×∫

Rpα1

((T (u′1 ∧ u′2))−1/α(x− y)

)ϕ(y)

×∫

Rpα1

((T |u′1 − u′2|)−1/α(y − z)

)ϕ(z)dzdydu′2du

′1

dx

|x|γ

=∫

R

∫ s1

0

∫ s2

0

(u′1 ∧ u′2)−γ/α|u′1 − u′2|−1/α

∫

Rpα1

((x′ − (T (u′1 ∧ u′2))−1/αy)

)ϕ(y)

×∫

Rpα1

((T |u′1 − u′2|)−1/α(y − z)

)ϕ(z)dzdydu′2du

′1

dx′

|x′|γ .(3.14)

Taking T →∞ in (3.14) we obtain the limit

pα1 (0)

(∫

Rϕ(y)dy

)2 ∫

R

pα1 (x)|x|γ dx

∫ s1

0

∫ s2

0

(u′1 ∧ u′2)−γ/α|u′1 − u′2|−1/αdu′2du′1

= pα1 (0)

(∫

Rϕ(y)dy

)2 ∫

R

pα1 (x)|x|γ dx

× 11− 1/α

∫ s1∧s2

0

u−γ/α[(s1 − u)1−1/α + (s2 − u)1−1/α

]du. (3.15)

By (3.13), (3.14) and (3.15),

limT→∞

1F 2

T

∫

R×RE(x1,x2)

( k∑

j=1

dj〈LTsj ,tj

;ϕ⊗ ψ〉)2

µ1(dx1)µ2(dx2)

=pα11 (0)

1− 1/α1

pα21 (0)

1− 1/α2

(∫

Rϕ(y)dy

)2(∫

Rψ(y)dy

)2 ∫

R

pα11 (x)|x|γ1

dx

∫

R

pα21 (x)|x|γ2

dx

10 JOHANNA GARZON

×k∑

j,j′=1

djd′j

∫ sj∧sj′

0

u−γ1/α1 [(sj − u)1−1/α1 + (sj′ − u)1−1/α1 ]du

×∫ tj∧tj′

0

v−γ2/α2 [(tj − v)1−1/α2 + (tj′ − v)1−1/α2 ]dv

= D2k∑

j,j′=1

djd′jC

(1)(sj , sj′)C(2)(tj , tj′), (3.16)

where D is defined by (3.7) and C(i) is as in (1.2) with ai = −γi/αi, bi = 1−1/αi.Proceeding similary with the third order term we find

∫

R×R

1F 3

T

E(x1,x2)

( k∑

j=1

dj〈Lsj ,tj, ϕ⊗ ψ〉

)3

µ1(dx1)µ2(dx2)

=1F 3

T

k∑

i=1

di

k∑

j=1

dj

k∑

l=1

dl

∫

R×RE(x1,x2)(〈LT

si,ti, ϕ⊗ ψ〉〈LT

sj ,tj, ϕ⊗ ψ〉

× 〈LTsl,tl

, ϕ⊗ ψ〉)µ1(dx1)µ2(dx2)

=1F 3

T

k∑

i=1

di

k∑

j=1

dj

k∑

l=1

dl

∫

R×R

∫ Tsi

0

∫ Tti

0

∫ Tsj

0

∫ Ttj

0

∫ Tsl

0

∫ Ttl

0

E(x1,x2)(〈Nui,vi , ϕ⊗ ψ〉〈Nu2,v2 , ϕ⊗ ψ〉〈Nu3,v3 , ϕ⊗ ψ〉)dv3du3dv2du2dv1du1µ1(dx1)µ2(dx2)

=k∑

i=1

di

k∑

j=1

dj

k∑

l=1

dl1

F(1)3T

∫

R

∫ Tsi

0

∫ Tsj

0

∫ Tsl

0

T α1u1ϕ(T α1

u2−u1ϕ(T α1

u3−u2ϕ))(x1)

du3du2du1dx1

|x1|γ1

× 1

F(2)3T

∫

R

∫ Tti

0

∫ Ttj

0

∫ Ttl

0

T α2v1ψ(T α2

v2−v1ψ(T α2

v3−v2ψ))(x2)dv3dv2dv1

dx2

|x2|γ2,

(3.17)

u1, u2, u3 denoting u1, u2, u3 in increasing order, and similarly for v1, v2, v3.Again, recalling (3.6), substituting ui = T u′i i = 1, 2, 3, using self-similarity of

the α-stable process, and then substituting x = (T u′1)1/α

x′, we have

1(T 1−(1+γ)/2α

)3

∫

R

∫ Ts1

0

∫ Ts2

0

∫ Ts3

0

T αu1ϕ(T α

u2−u1ϕ(T α

u3−u2ϕ))(x)du3du2du1

dx

|x|γ

= T (γ−1)/2α

∫

R

∫ s1

0

∫ s2

0

∫ s3

0

∫

Ru−γ/α1 (u2 − u1)−1/α(u3 − u2)−1/α

×∫

Rpα1 (x′ − (T u1)−1/αw)ϕ(w)

∫

Rpα1 ((T (u2 − u1))−1/α(w − y))ϕ(y)

×∫

Rpα1 ((T (u3 − u2))−1/α(y − z))ϕ(z)dzdydwdu3du2du1

dx′

|x′|γ , (3.18)


then from (3.17) and (3.18),

1F 3

T

∫

R×RE(x1,x2)

( k∑

j=1

dj〈LTsj ,tj

, ϕ⊗ ψ〉)3

µ1(dx1)µ2(dx2)

=2∏

i=1

T (γi−1)/2αi

∫

R×RAT (x1, x2)µ1(dx1)µ2(dx2), (3.19)

where

AT (x1, x2) =∫ s1

0

∫ s2

0

∫ s3

0

u−γ1/α11 (u2 − u1)−1/α1(u3 − u2)−1/α1

×∫

Rpα11 (x1 − (T u1)−1/α1w)ϕ(w)

∫

Rpα11 ((T (u2 − u1))−1/α1(w − y))ϕ(y)

×∫

Rpα11 ((T (u3 − u2))−1/α1(y − z))ϕ(z)dzdydwdu3du2du1

×∫ t1

0

∫ t2

0

∫ t3

0

∫

Rv−γ2/α21 (v2 − v1)−1/α2(v3 − v2)−1/α2

×∫

Rpα21 (x2 − (T v1)−1/α2w)ψ(w)

∫

Rpα21 ((T (v2 − v1))−1/α2(w − y))ψ(y)

×∫

Rpα21 ((T (v3 − v2))−1/α2(y − z))ψ(z)dzdydwdv3dv2dv1. (3.20)

From (3.20) we obtain

limT→∞

∫

R×RAT (x1, x2)µ1(dx1)µ2(dx2)

=∫ s1

0

∫ s2

0

∫ s3

0

u−γ1/α11 (u2 − u1)−1/α1(u3 − u2)−1/α1du3du2du1

×∫ t1

0

∫ t2

0

∫ t3

0

v−γ2/α21 (v2 − v1)−1/α2(v3 − v2)−1/α2dv3dv2dv1

×(∫

Rϕ(x)dx

)3 (∫

Rψ(x)dx

)3 2∏

i=1

(pαi1 (0))2

∫

R

pαi1 (x)|x|γi

dx. (3.21)

Then, from (3.11), (3.19) and (3.21) we get

limT→∞

∫

R×RδT(x1,x2)

µ1(dx1)µ2(dx2) = 0. (3.22)

Finally, putting (3.12), (3.16) and (3.22) together we obtain

limT→∞

CT (θ) = exp−θ

2

2D2

k∑

j=1

k∑

j′=1

djdj′C(1)(sj , sj′)C(2)(tj , tj′)

,

and convergence of finite-dimensional distributions of XT to finite-dimensionaldistributions of weighted fractional Brownian sheet DW has been proved. ¤

Theorem 3.2. Under the hypotheses of Theorem 3.1, if γ1 = γ2 = 0, then XT

converges in law to DW in the space of continuous functions C([0, τ ] × [0, τ ],R)

12 JOHANNA GARZON

for any τ > 0 as T → ∞, where W is fractional Brownian sheet with parameters(1− 1

2α1, 1− 1

2α2), and

D =∫

Rϕ(x)dx

∫

Rψ(x)dx

[ 2∏

i=1

11− 1/αi

pαii (0)

]1/2

.

Proof. By Theorem 3.1 we have convergence of finite-dimensional distributions ofXT to those of DW . It remains to show that the family XT is tight. Sincethese processes are null on the axes, by the Bickel-Wichura theorem [4] we onlyneed prove that there exist even m ≥ 2 and positive constants Cm, δ1, δ2 such thatmδ1,mδ2 > 1 and

supTE ((∆s1,t1XT (s2, t2))

m) ≤ Cm(s2−s1)mδ1(t2− t1)mδ2 , for all s1 < s2, t1 < t2.

(3.23)From (3.4),

∆s1,t1XT (s2, t2) =1FT

∫ Ts2

Ts1

∫ Tt2

Tt1

(〈Nu,v, ϕ⊗ ψ〉 − E(〈Nu,v, ϕ⊗ ψ〉)) dudv,

then, by (3.10),

E((∆s1,t1XT (s2, t2))2)

=1F 2

T

∫ Ts2

Ts1

∫ Tt2

Tt1

∫ Ts2

Ts1

∫ Tt2

Tt1

Cov (〈Nu1,v1 , ϕ⊗ ψ〉 , 〈Nu2,v2 , ϕ⊗ ψ〉) dv2du2dv1du1

=1F 2

T

∫ Ts2

Ts1

∫ Tt2

Tt1

∫ Ts2

Ts1

∫ Tt2

Tt1

∫

RT α1

u1∧u2(ϕT α1

|u1−u2|ϕ)(x1)dx1

×∫

RT α2

v1∧v2(ϕT α2

|v1−v2|ϕ)(x2)dx2dv2du2dv1du1

=1

T 2−(1+γ1)/α1

∫ Ts2

Ts1

∫ Ts2

Ts1

∫

RT α1

u1∧u2(ϕT α1

|u1−u2|ϕ)(x1)dx1du2du1

× 1T 2−(1+γ2)/α2

∫ Tt2

Tt1

∫ Tt2

Tt1

∫

RT α2

v1∧v2(ϕT α2

|v1−v2|ϕ)(x2)dx2dv2dv1

= E(〈X(1)T (s2)−X

(1)T (s1), ϕ〉2)E(〈X(2)

T (t2)−X(2)T (t1), ψ〉2), (3.24)

where X(i)T , i = 1, 2, are occupation time fluctuation processes of independent

systems of particles moving in R according to symmetric αi-stable processes withinitial configurations given by Poisson random measures on R with intensities µi,i.e.,

X(1)T (t) =

1T 1−1/2α1

∫ Tt

0

(〈N (1)u , ϕ〉 − E(〈N (1)

u , ϕ〉))du, N (1)u =

∑

x∈Pois(µ1)

δξxu,

and

X(2)T (t) =

1T 1−1/2α2

∫ Tt

0

(〈N (2)u , ϕ〉 − E(〈N (2)

u , ϕ〉))du, N (2)u =

∑

x∈Pois(µ2)

δζxu.


In [6] such a one-time system is studied and it is shown that

E(〈X(i)T (t)−X

(i)T (s), ϕ〉2) ≤ C|t− s|h, (3.25)

where C is a positive constant (not depending on T ) and h = 2 − 1αi> 1. From

(3.24) and (3.25) we obtain (3.23). ¤

Remark 3.3. We make some remarks below.

(1) Theorem 3.2 gives a functional approximation of fractional Brownian sheetwith parameters (h1, h2) ∈ (1/2, 3/4]2, taking hi = 1− 1

2αi, i = 1, 2.

(2) Proving tightness with γ1 6= 0 or γ2 6= 0 is considerably more difficultbecause it requires computing moments of arbitrarily high order (see [8]for the one time case), and this involves moments of arbitrarily high orderof the Poisson random measure Pois(µ1 ⊗ µ2), which are cumbersome.

(3) In Theorem 3.1 we may also consider the measures µi of the form (3.5)with γi < 0, assuming that |γi| < αi if αi < 2 (which implies that themean is finite), and the result in the theorem holds.

(4) The role of the functions ϕ,ψ is only subsidiary since they are fixed, andin the occupation time fluctuation limit they appear only in the constantD given by (3.7). If ϕ,ψ are taken as variables in the space S(R) ofsmooth rapidly decreasing functions, then in principle it is possible toprove convergence of the occupation time fluctuations as (S ′(R))2-valuedprocesses, where S ′(R) is the space of tempered distributions (topologicaldual of S(R)), the limit being the space-time random field

(Zs,t)s,t≥0 = K(λ⊗ λ) (Ws,t)s,t≥0 ,

where W is the weighted fractional Brownian sheet in Theorem 3.1,

K =

(2∏

i=1

11− 1/αi

pαi1 (0)

∫

R

pαi1 (x)|x|γi

dx

)1/2

,

and λ is the Lebesgue measure on R. (See [8] for such a setup for aone-time particle system.)

References

1. Ayache, A., Leger S., and Pontier, M.: Drap brownien fractionnaire, Potential Anal. 17(2002) 31–43.

2. Bardina, X., Jolis, M., and Tudor, C. A.: Weak convergence to the fractional Brownian sheetand other two-parameter Gaussian processes, Stat. Probab. Lett. 65 (2003) 317–329.

3. Bardina, X. and Florit, C.: Approximation in law to the d−parameter fractional Browniansheet based on the functional invariance principle, Rev. Mat. Iberoamericana 21 (2005) 1037–1052.

4. Bickel, P. and Wichura, M.: Convergence criteria for multiparameter stochastic processesand some applications, Ann. Math. Statist. 42 (1971) 1656–1270.

5. Billingsley, P.: Probability and Measure, Wiley, 1979.6. Bojdecki, T., Gorostiza, L. G., and Talarczyk, A.: Limits theorems for occupation time

fluctuations of branching systems: I Long-range dependence, Stoch. Proc. Appl. 116 (2006)1–18.

14 JOHANNA GARZON

7. Bojdecki, T., Gorostiza, L. G., and Talarczyk, A.: Some extensions of fractional Brownianmotion and sub-fractional Brownian motion related to particle systems, Elect. Comm. inProbab. 12 (2007) 161–172.

8. Bojdecki, T., Gorostiza, L. G., and Talarczyk, A.: Occupation time limits of inhomogeneousPoisson systems of independent particles, Stoch. Proc. Appl. 118 (2008) 28–52.

9. Coutin, L. and Pointer, M.: Approximation of the fractional Brownian sheet via Ornstein-Uhlenbeck sheet, ESAIM PS 11 (2007) 115–146.

10. Feyel, D. and De La Pradelle, A.: Fractional integrals and Brownian processes, PotentialAnal. 10 (1991) 273–288.

11. Kallenberg, O.: Random Measures, Akademie-Verlag, Berlin (1986).12. Kamont, A.: On the fractional anisotropic Wiener field, Prob. Math. Statist. 16 (1996)

85–98.13. Kuhn, T. and Linde, W.: Optimal series representations of fractional Brownian sheets,

Bernoulli 8 (2002) 669–696.14. Nzi, M. and Mendy, I.: Approximation of fractional Brownian sheet by a random walk in

anisotropic Besov space, Random Oper. Stoch. Equ. 15 (2007) 137–154.15. Tudor, C. A.: Weak convergence to the fractional Brownian sheet in Besov spaces, Bulletin

of the Brazilian Mathematical Society 34(3) (2003) 389-400.

Johanna Garzon: Department of Mathematics, CINVESTAV, Mexico City, MexicoE-mail address: [email protected]

PAC COMMUTATORS AND THE R–TRANSFORM

AUREL I. STAN

Abstract. We develop an algorithmic method for working out moments ofa probability measure on the real line from the preservation–annihilation–creation operator commutator relations for the measure. The method is ap-plied to prove a result of Voiculescu on the R–transform.

1. Introduction

Expanding on a program introduced in [1] and continued in [2], it was provenin [8] that the moments of a probability distribution can be recovered from thecommutator between its annihilation and creation operators, and the commutatorbetween its annihilation and preservation operators, provided that the first ordermoment is given. Moreover, a simple, concrete method for computing the momentswas introduced in [8]. In the present paper we apply this method to some classicdistributions and to give another proof of a theorem of Voiculescu concerningthe R–transform. There are already some known techniques for recovering themoments or even the probability distribution of a random variable X, havingfinite moments of all orders, from its Szego–Jacobi parameters. One method usesa continued fraction expansion of the Cauchy–Stieltjes transform of X and is veryuseful when the random variable has a compact support. Another powerful wayis the method of renormalization introduced in [3, 5, 4] and pushed almost to thelimits in [7]. However, our method is based on the Lie algebra structure of thealgebra generated by the annihilation, preservation, and creation operators.

In section 2 we introduce very quickly the annihilation, preservation, and cre-ation operators for a one–dimensional distribution having finite moments of anyorder. For brevity we will call these operators the PAC operators. We also presentthe commutator method and its dual developed in [8]. In section 3 we apply thismethod to two families of distributions. Finally, in section 4, we use the commu-tator method and its dual to give a proof of an important theorem, of Voiculescu,about the analytic function theory tools for computing the R–transform.

2. Background

Let X be a random variable having finite moments of any order, i.e., E[|X|p] <∞, for all p > 0, where E denotes the expectation. It is well–known that by apply-ing the Gram–Schmidt orthogonalization procedure to the sequence of monomial

2000 Mathematics Subject Classification. Primary 81S25; Secondary 05E35.Key words and phrases. Moments, Szego–Jacobi parameters, creation, annihilation, preser-

vation, commutator, Cauchy–Stieltjes transform, R–transform.

15



16 AUREL I. STAN

random variables: 1, X, X2, . . . , we can obtain a sequence of orthogonal poly-nomial random variables: f0(X), f1(X), f2(X), . . . , chosen such that for eachn ≥ 0, fn has degree n and a leading coefficient equal to 1. We assume that theprobability distribution of X has an infinite support so that the sequence f0, f1,fn, . . . never terminates. There exist two sequences of real numbers: αkk≥0

and ωkk≥1, called the Szego–Jacobi parameters of X, such that for all n ≥ 0, wehave:

Xfn(X) = fn+1(X) + αnfn(X) + ωnfn−1(X). (2.1)

See [6] and [9]. When n = 0, in this recursive relation f−1 = 0 (the null polynomial)and ω0 := 0 by agreement. The terms of the sequence ωkk≥1 are called theprincipal Szego–Jacobi parameters of X and they must all be positive since, forall n ≥ 1,

E[fn(X)2

]= ω1ω2 · · ·ωn. (2.2)

Moreover, by Favard’s theorem, given any sequence of real numbers αkk≥0 andany sequence of positive numbers ωkk≥1, there exists a random variable X havingthese sequences as its Szego–Jacobi parameters.

Let F be the space of all random variables of the form f(X), where f is apolynomial function, and for each n ≥ 0, let Fn be the subspace of F consisting ofall random variables f(X), such that f is a polynomial of degree at most n. LetG0 := F0, and for all n ≥ 1, let Gn := FnªFn−1, i.e., the orthogonal complementof Fn−1 into Fn. For each n ≥ 0, Gn = Cfn(X) (scalar multiples of fn(X)) andGn is called the homogenous chaos space of degree n generated by X. The spaceH := ⊕n≥0Gn is called the chaos space generated by X. It is clear that F is densein H. For each n ≥ 0, we denote by Pn the orthogonal projection of H onto Gn.If we look back to the recursive formula (2.1), then we can easily see that, for alln ≥ 0:

Pn+1 [Xfn(X)] = fn+1(X),

Pn [Xfn(X)] = αnfn(X),

and

Pn−1 [Xfn(X)] = ωnfn−1(X).

Let us regard now X not as a random variable, but as a multiplication operatorfrom F to F , which maps a polynomial random variable f(X) into Xf(X). Wecan see that applying the multiplication operator X to a polynomial from Gn weget three polynomials: one in Gn+1, one in Gn, and one in Gn−1. That means:

X|Gn = Pn+1X|Gn + PnX|Gn + Pn−1X|Gn.

We define D+n := Pn+1X|Gn, D0

n := PnX|Gn, and D−n := Pn−1X|Gn. Since

D+n maps Gn into Gn+1, it increases the degree of fn, and thus it is called a

creation operator. Similarly, since D0n : Gn → Gn, it is called a preservation

operator, and since D−n : Gn → Gn−1, it is called an annihilation operator. So far

the annihilation, preservation, and creation operators have been defined only oneach individual homogenous chaos space Gn. We extend their definition as linearoperators from F to F , and define the operators: a−, a0, and a+, such that for

PAC COMMUTATORS AND THE R–TRANSFORM 17

any n ≥ 0, a−|Gn := D−n , a0|Gn := D0

n, and a+|Gn := D+n . As a multiplication

operator X is the sum of these three operators. Thus:

X = a− + a0 + a+. (2.3)

It is known that X is polynomially symmetric, i.e., E[X2k−1] = 0, for all positiveintegers k, if and only if a0 = 0, see [1], or equivalently αn = 0, for all n ≥ 0. Inthis case X = a− + a+.

We will briefly explain now the commutator method introduced in [8], usedto recover the moments and if possible the probability distribution of a randomvariable X, from the commutator between its annihilation and creation operators,commutator between its annihilation and preservation operators, and its first mo-ment, E[X]. We would like to make the reader aware of the fact that some timeswe regard X as a random variable, and other times we view it as a multiplicationoperator. We hope that this will not create any confusion, since most of the time,when we refer to it as being a multiplication operator, we will write Xn1, where 1is the constant (vacuum) polynomial equal to 1. When we write E[Xn], for somen ≥ 1, we regard X as a random variable. The commutator of two operators Aand B is defined as:

[A,B] := AB −BA. (2.4)

Commutator MethodLet X be a random variable, having finite moments of all orders. We assume

that [a−, a+], [a−, a0], and E[X] are given (known). Then, in order to computethe higher moments of X, we will follow the following three steps.Step 1. Let 〈·, ·〉 denote the inner product defined as:

〈f(X), g(X)〉 := E[f(X)g(X)],

for all polynomials f and g. For any fixed positive integer n, we have:

E[Xn] = 〈XXn−11, 1〉= 〈(a− + a0 + a+)Xn−11, 1〉= 〈a−Xn−11, 1〉+ 〈a0Xn−11, 1〉+ 〈a+Xn−11, 1〉.

Since (a0)∗ = a0, (a+)∗ = a−, a01 = E[X]1, and a−1 = 0, we have:

〈a+Xn−11, 1〉 = 〈Xn−11, a−1〉= 〈Xn−11, 0〉= 0

and

〈a0Xn−11, 1〉 = 〈Xn−11, a01〉= 〈Xn−11, E[X]1〉= E[X]〈Xn−11, 1〉= E[X]E[Xn−1].

Thus

E[Xn] = E[X]E[Xn−1] + 〈a−Xn−11, 1〉.

18 AUREL I. STAN

Step 2. Swap (permute) a− and Xn−1, using the simple formula:

AB = BA + [A,B]

and the product rule for commutators:

[A,Bk] =k−1∑

j=0

Bk−1−j [A,B]Bj ,

for all operators A and B, and any k ≥ 2. Use also the fact that

[a−, X] = [a−, a− + a0 + a+]= [a−, a−] + [a−, a0] + [a−, a+]= [a−, a0] + [a−, a+].

Thus we get:

E[Xn] = E[X]E[Xn−1] + 〈a−Xn−11, 1〉= E[X]E[Xn−1] + 〈(Xn−1a− + [a−, Xn−1]

)1, 1〉

= E[X]E[Xn−1] + 〈Xn−1a−1, 1〉+ 〈[a−, Xn−1]1, 1〉= E[X]E[Xn−1] + 〈[a−, Xn−1]1, 1〉

= E[X]E[Xn−1] +n−2∑

j=0

〈Xn−2−j [a−, X]Xj1, 1〉

= E[X]E[Xn−1] +n−2∑

j=0

〈Xn−2−j([a−, a0] + [a−, a+]

)Xj1, 1〉.

Step 3. If necessary, go back to Step 2 and repeat the procedure, until a recursiveformula expressing the n–th moment in terms of lower order moments is obtained.

The idea in this method (algorithm) is very simple: move each annihilator a−

stepwise to the right, using the commutator relations with a+ and a0, until it actson the vacuum polynomial 1 and kills it. There is also a dual of this method, usingthe creation operator a+, instead of the annihilation operator a−. We will brieflyexplain it now.Dual Commutator MethodStep 1. For any fixed positive integer n, we have:

E[Xn] = 〈Xn−1X1, 1〉= 〈Xn−1(a− + a0 + a+)1, 1〉= 〈Xn−1a−1, 1〉+ 〈Xn−1a01, 1〉+ 〈Xn−1a+1, 1〉= 0 + 〈Xn−1E[X]1, 1〉+ 〈Xn−1a+1, 1〉= E[X]E[Xn−1] + 〈Xn−1a+1, 1〉.

Step 2. Swap Xn−1 and a+.Step 3. Repeat Step 2 if necessary.

In the dual commutator method the creation operator a+ is moved stepwise tothe left, until it arrives to the left most possible position. In that moment, for any


polynomial f , we have:

〈a+f(X)1, 1〉 = 〈f(X)1, a−1〉= 0.

3. Some Calculations

In this section we apply the commutator method to two concrete examples.

Example 3.1. Let us consider now a random variable X, having finite momentsof all orders, whose Szego–Jacobi parameters are αn = 0, for all n ≥ 0, and theprincipal Szego–Jacobi parameters are: c, c + d, 2c + d, 2c + 2d, 3c + 2d, 3c + 3d,. . . . That means, for all n ≥ 1, ω2n−1 = nc + (n− 1)d and ω2n = nc + nd, wherec and d are fixed real numbers, such that c > 0 and c + d > 0.

Since αn = 0, for all n ≥ 0, we know that X is symmetric and thus the spacespanned by the monomial random variables of even degree: 1, X2, X4, . . . , isorthogonal to the space spanned by the monomial random variables of odd degrees:X, X3, X5, . . . . In fact the closures of these two spaces areHe := G0⊕G2⊕G4⊕· · ·and Ho := G1 ⊕ G3 ⊕ G5 ⊕ · · · . Let E : H → He and O : H → Ho denote theorthogonal projections of H onto He and Ho, respectively. Since, for all n ≥ 0,[a−, a+]fn(X) = (ωn+1−ωn)fn(X), where fnn≥0 are the orthogonal polynomialsgenerated by X, we can see that the commutator of the annihilation and creationoperators is:

[a−, a+] = cE + dO. (3.1)

Because a0 = 0, all the odd moments vanish. Applying now our commutatormethod, for all n ≥ 1, we have:

E[X2n] =2n−2∑

j=0

〈X2n−2−j [a−, a+]Xj1, 1〉

= c2n−2∑

j=0

〈X2n−2−jEXj1, 1〉+ d2n−2∑

j=0

〈X2n−2−jOXj1, 1〉.

Since EX2k = X2k, EX2k+1 = 0, OX2k+1 = X2k+1, and OX2k = 0, for all k ≥ 0,we get:

E[X2n] = cn−1∑

k=0

〈X2n−2−2kX2k1, 1〉+ dn−2∑

k=0

〈X2n−2−2k−1X2k+11, 1〉

= cnE[X2n−2] + d(n− 1)E[X2n−2]= [(c + d)n− d]E[X2n−2].

Iterating this recursive relation, we obtain:

E[X2n] = [(c + d)n− d]E[X2n−2]= [(c + d)n− d][(c + d)(n− 1)− d]E[X2n−4]· · · . . .

= [(c + d)n− d][(c + d)(n− 1)− d] · · · [(c + d)1− d]E[X0].

20 AUREL I. STAN

Thus we obtain that:

1(c + d)n

E[X2n] =(

n− d

c + d

) (n− 1− d

c + d

). . .

(1− d

c + d

). (3.2)

We recognize that the right–hand side of (3.2) is exactly the n–th moment of agamma distribution. That means the distribution of the random variable Y :=[1/(c + d)]X2 is given by the density function:

f(x) =1

Γ(

cc+d

)xc

c+d−1e−x1(0,∞)(x), (3.3)

where Γ denotes the Euler’s gamma function. Thus X2 is a re–scaled gammarandom variable. Since X is a symmetric random variable, we can compute firstits distribution function FX in the following way:

FX(a) := P (X ≤ a)= 1− P (X > a)

= 1− 12P (X2 > a2)

=12

+12P ([1/(c + d)]X2 ≤ a2/(c + d))

=12

+12FY (a2/(c + d)),

for all a > 0. Differentiating both sides of the last equality with respect to a, weobtain that the density of X is:

g(a) = F ′X(a)

=a

c + dF ′Y (a2/(c + d))

=a

c + df(a2/(c + d)),

for all a > 0. Since g(−a) = g(a), we conclude that X is the random variablegiven by the density function:

g(x) =1

(c + d)c

c+d Γ(

cc+d

) |x| c−dc+d e−

x2c+d . (3.4)

Example 3.2. Let us find now the random variable X whose Szego–Jacobi pa-rameters are:

αn =

α if n = 00 if n ≥ 1

and

ωn =

b if n = 1c if n ≥ 2,

where α, b, and c are fixed real numbers, such that b and c are strictly positive.


Before computing the moments of X, we will find a simple upper bound forE[|(X − α)n|], for each n ≥ 0.Claim 1. For each n ≥ 0, we have:

E[|(X − α)n|] ≤ (3T )n, (3.5)

where T := max1, |α|, b, c.Indeed, let n ≥ 0 be fixed. Let fnn≥0 denote the sequence of orthogonal

polynomials, with a leading coefficient equal to 1, generated by X. We have:

E[(X − α)2n] = 〈(X − αI) · · · (X − αI)1, 1〉= 〈a− + a+ + (a0 − αI) · · · a− + a+ + (a0 − αI)1, 1〉=

∑

(a1,··· ,a2n)∈a−,a+,a0−αI2n

〈a1 · · · a2n1, 1〉.

Observe that in the last sum only the terms that contain the same number ofannihilation and creation operators could be non–zero, since we start from thevacuum space R1 and we have to return to this space (otherwise a1 · · · a2n1⊥1). Forthese terms, we move from one orthogonal polynomial to another in the followingway. If aj = a+, and we are currently at fk, then ajfk = fk+1, and we retaina coefficient cj = 1. If aj = a−, then ajfk = ωkfk−1 and we retain a coefficientcj = ωk. Finally, if aj = a0−αI, then ajfk = (αk−α)fk and we retain a coefficientcj = αk − α. Observe, that for all j, we have |cj | ≤ T . Since at the end we returnto the vacuum polynomial 1, we get:

E[(X − α)2n] =∑

〈c1 · · · c2n1, 1〉=

∑c1 · · · c2n

≤∑

|c1 · · · c2n|≤ 32nT 2n,

since the cardinality of the set a−, a0, a0 − αI2n is 32n. Using now Jensen’sinequality we get:

E[|X|n] ≤√

E[X2n]≤ (3T )n.

Let us compute now the moments of X. We have:

[a−, a+]fn = (ωn+1 − ωn)fn. (3.6)

Thus [a−, a+]f0 = bf0, [a−, a+]f1 = (c− b)f1, and [a−, a+]fn = 0, for all n ≥ 2.This means that:

[a−, a+] = bP0 + (c− b)P1, (3.7)

where Pk denotes the projection onto the space Gk = Cfk. Moreover, since

[a−, a0]fn = (αn − αn−1)ωnfn−1, (3.8)

for all n ≥ 0, where α−1 := 0, we conclude that:

[a−, a0] = −αa−P1. (3.9)

22 AUREL I. STAN

From the recursive relation:

Xf0(X) = f1(X) + α0f0(X) + ω0f−1(X),

since f0 = 1, we conclude that f1(X) = X − α. Moreover E[f1(X)2] = ω1 = b.Thus 1 and (1/

√b)(X−α) are orthonormal bases of G0 and G1, respectively.

Hence for all polynomial functions g, we have:

P0g(X) = 〈g(X), 1〉1= E[g(X)]1

and

P1g(X) = 〈g(X), (1/√

b)(X − α)〉(1/√

b)(X − α)

=1b〈g(X), X − α〉(X − α)

=1bE[(X − α)g(X)](X − α).

We apply now our commutator method to compute the moments of X. Actually,it is easier to compute the moments of X − α than those of X. For any fixednatural number n, we have:

E[(X − α)n] = 〈(a+ + a0 + a− − αI)(X − αI)n−11, 1〉= 〈a+(X − αI)n−11, 1〉+ 〈(a0 − αI)(X − αI)n−11, 1〉+ 〈a−(X − αI)n−11, 1〉.

Here I denotes the identity operator of H. We have:

〈a+(X − αI)n−11, 1〉 = 〈(X − αI)n−11, a−1〉= 0

and

〈(a0 − αI)(X − αI)n−11, 1〉 = 〈(X − αI)n−11, (a0 − αI)1〉= 0.

Thus we have:

E[(X − α)n] = 〈a−(X − αI)n−11, 1〉.

We swap now a− and (X − αI)n. Since after the swap the annihilation operatora− kills the vacuum polynomial 1, we get:

E[(X − α)n] = 〈[a−, (X − αI)n−1]1, 1〉.


Thus we obtain:

E[(X − α)n] = 〈[a−, (X − αI)n−1]1, 1〉

=n−2∑

j=0

〈(X − αI)n−2−j [a−, X − αI](X − αI)j1, 1〉

=n−2∑

j=0

〈(X − αI)n−2−j [a−, a+ + a0 + a− − αI](X − αI)j1, 1〉

=n−2∑

j=0

〈(X − αI)n−2−j [a−, a+](X − αI)j1, 1〉

+n−2∑

j=0

〈(X − αI)n−2−j [a−, a0](X − αI)j1, 1〉

=n−2∑

j=0

〈(X − αI)n−2−j [bP0 + (c− b)P1](X − αI)j1, 1〉

− αn−2∑

j=0

〈(X − αI)n−2−ja−P1(X − αI)j1, 1〉.

Since P0(X − α)j = E[(X − α)j ]1 and P1(X − α)j = (1/b)E[(X − α)j+1](X − α),we obtain the following recursive formula:

E[(X − α)n] = bn−2∑

j=0

E[(X − α)j ]E[(X − α)n−2−j ]

+c− b

b

n−2∑

j=0

E[(X − α)j+1]E[(X − α)n−1−j ]

− αn−2∑

j=0

E[(X − α)j+1]E[(X − α)n−2−j ],

for all n ≥ 1. Multiplying both sides of this recursive relation by tn and thensumming up from n = 1 to infinity, we obtain that the function ϕ(t) = E[1/(1 −t(X − α))] satisfies the following equation:

ϕ(t)− 1 = bt2ϕ2(t) +c− b

b[ϕ(t)− 1]2 − αtϕ(t)[ϕ(t)− 1],

for all t in a neighborhood of 0. It must be observed, that in deriving this formulawe interchanged the summation with the expectation, which is possible for thesmall values of t, due to the inequality (3.5). This relation is equivalent to thequadratic equation in ϕ(t):

(bt2 − αt + p− 1)ϕ2(t) + (αt− 2p + 1)ϕ(t) + p = 0, (3.10)

24 AUREL I. STAN

where p := c/b. Using the quadratic formula, we get:

ϕ(t) =−αt + 2p− 1±

√(αt + 1)2 − 4pbt2

2(bt2 − αt + p− 1)

=−αt + 2p− 1±

√(αt + 1)2 − 4pbt2

2(bt2 − αt + p− 1)

× −αt + 2p− 1∓√

(αt + 1)2 − 4pbt2

−αt + 2p− 1∓√

(αt + 1)2 − 4pbt2

=4p

(bt2 − αt + p− 1

)

2 (bt2 − αt + p− 1)(−αt + 2p− 1∓

√(αt + 1)2 − 4pbt2

)

=2p

−αt + 2p− 1∓√

(αt + 1)2 − 4pbt2.

Since, ϕ(0) = E[1] = 1, we get:

ϕ(t) =2p

−αt + 2p− 1 +√

(αt + 1)2 − 4pbt2,

for all t in a neighborhood of 0. Thus we get

E

[1

1− t(X − α)

]=

2p

−αt + 2p− 1 +√

(αt + 1)2 − 4pbt2. (3.11)

Replacing t by 1/t, we obtain that the Cauchy–Stieltjes transform of X is:

E

[1

t− (X − α)

]=

2p

−α + (2p− 1)t + s(t)√

(t + α)2 − 4pb, (3.12)

for all t away from 0, where s(t) denotes the sign function of t, i.e., s(t) = t/|t|.We can invert the Cauchy–Stieltjes transform to find the probability distribution

of X − α first, and then of X. We are not going over this computation, but theinterested reader can read Theorem 5.3 from [7].

4. The R–transform

We will close the paper, by giving a new proof of a theorem by Voiculescuconcerning the analytic function theory tools for computing the R–transform. Wewill briefly explain this transform, following the concepts from [10].

Let H = Ce be a one–dimensional Hilbert space, where e is an orthonormalbasis of H. Let Γ(H) be the full Fock space generated by H, that means:

Γ(H) := C1⊕H ⊕H⊗2 ⊕H⊗3 ⊕ · · · ,

where “⊕” means orthogonal direct sum. We define a left creation operator a+

(denoted by l in [10]) on Γ(H) in the following way:

a+τ =

e if τ = 1e⊗ τ if τ ∈ Γ(H)ª C1,

(4.1)


where “ª” denotes the orthogonal complement. The adjoint of this operator isthe left annihilation operator a− (denoted by l∗ in [10]) and is defined as:

a−(k1 ⊗ k2 ⊗ · · · ⊗ kn) = 〈k1, e〉k2 ⊗ · · · ⊗ kn if n ≥ 1

0 if n = 0,(4.2)

where for n = 0, k1 ⊗ k2 ⊗ · · · ⊗ kn is understood to be a complex multiple of 1(that means an element of C1), and 〈·, ·〉 denotes the inner product of H.

It is not hard to see that the commutator of the left creation and annihilationoperators is:

[a−, a+] = P0, (4.3)

where P0 denotes the orthogonal projection of Γ(H) onto the vacuum space C1.Moreover, a−a+ = I, where I denotes the identity operator of Γ(H).

Definition 4.1. A noncommutative probability space is a unital algebra, A overC together with a linear functional, φ : A → C, such that φ(1) = 1.

Every element f in A is called a random variable, and φ(f) is called the expec-tation of f . For this reason we will replace the letter “φ” from [10] by “E”. Everyrandom variable f from A generates a distribution µf on the algebra of complexpolynomials in one variable C[X], i.e., a linear functional from C[X] to C thatmaps the constant polynomial 1 into the complex number 1. It is defined by theformula:

µf (P [X]) := E[P (f)], (4.4)

for all P [X] ∈ C[X]. Let Σ denote the set of all linear functionals µ, on C[X],such that µ(1) = 1.

Proposition 4.2. For all p1, p2, q1, and q2 non-negative integers, we have:

(a+)p1(a−)q1 = (a+)p2(a−)q2 (4.5)

if and only if p1 = p2 and q1 = q2.

Proof. Let us assume that (a+)p1(a−)q1 = (a+)p2(a−)q2 . Since for all k ≥ q1,(a+)p1(a−)q1 maps H⊗k into H⊗(k+p1−q1), and for all k ≥ q2, (a+)p2(a−)q2 mapsH⊗k into H⊗(k+p2−q2), we conclude that:

p1 − q1 = p2 − q2. (4.6)

Let us assume that p2 ≥ p1 and thus, m := p2−p1 = q2−q1 ≥ 0. By composing(a−)p1 with each side of the equality (4.5), we get:

(a−)p1(a+)p1(a−)q1 = (a−)p1(a+)p2(a−)q2 .

Since a−a+ = I and p2 ≥ p1, we obtain:

(a−)q1 = (a+)p2−p1(a−)q2 ,

which means:

(a−)q1 = (a+)m(a−)q2 . (4.7)

26 AUREL I. STAN

Let us compose now each side of the equality (4.7) with (a+)q1 (to the right). Weobtain:

(a−)q1(a+)q1 = (a+)m(a−)q2(a+)q1 .

Since a−a+ = I and q2 ≥ q1, we have:

I = (a+)m(a−)q2−q1 ,

which means:

I = (a+)m(a−)m. (4.8)

If m > 0, then the equality (4.8) is impossible since if we apply each side of it tothe vacuum vector 1, we get: I1 = 1 while (a+)m(a−)m1 = 0, because a− kills thevacuum vector. Thus m = 0 and so, p1 = p2 and q1 = q2. ¤

Proposition 4.2 allows us to define the unital algebra E1 of formal series of theform:

Q∑q=0

∞∑p=0

cp,q(a+)p(a−)q, (4.9)

where Q ≥ 0 and cp,q ∈ C, for all 0 ≤ q ≤ Q and p ≥ 0.E1 is an algebra (i.e., closed under multiplication) due to the fact that the

creation operators are always to the left of the annihilation operators, and a−a+ =I. We define the expectation E (i.e., the linear functional φ mapping 1 to 1), onE , in the following way:

E

[Q∑

q=0

∞∑p=0

cp,q(a+)p(a−)q

]:= c0,0. (4.10)

Let us observe that formally, for any f ∈ E1, we have:

E[f ] = 〈〈f1, 1〉〉, (4.11)

where 〈〈·, ·〉〉 denotes the inner product of the Fock space Γ(H) and 1 the vacuumvector of Γ(H). Thus (E , E) is a noncommutative probability space.

Voiculescu proved (see [10]) that, for every µ ∈ Σ (let us remember that Σdenotes the set of all linear functionals µ, on C[X], such that µ(1) = 1), thereexists a unique random variable, Tµ, of the form a− +

∑∞k=0 αk+1(a+)k in E1,

whose distribution in (E , E) is µ. Here the numbers α1, α2, . . . , represent arbitrarycoefficients and have nothing to do with the Szego–Jacobi parameters. Tµ is calledthe canonical random variable of µ. We define the R–transform of µ to be theformal power series:

Rµ =∞∑

k=0

αk+1xk. (4.12)

We will give a proof of the following theorem (Theorem 3.3.1. from [10]), usingboth our commutator and dual commutator method.


Theorem 4.3. Let µ be a distribution on C[X], with R–transform

Rµ(z) =∞∑

k=0

αk+1zk. (4.13)

Then denoting by µk the kth moment of µ, µ(Xk), we have that the formal powerseries

G(w) = w−1 +∞∑

k=1

µkw−k−1 (4.14)

and

K(z) =1z

+Rµ(z) (4.15)

are inverses with respect to composition.

Proof. Let Tµ := a−+∑∞

k=0 αk+1(a+)k ∈ E1 be the canonical random variable ofµ. Let us compute the moments of µ, or equivalently of Tµ, using our commutatormethod. For all n ≥ 1, we have:

µn = E[Tn

µ

]

= 〈〈Tnµ 1, 1〉〉

= 〈〈[a− +

∞∑

k=0

αk+1(a+)k

]Tn−1

µ 1, 1〉〉

= 〈〈a−Tn−1µ 1, 1〉〉+ α1〈〈Tn−1

µ 1, 1〉〉+∞∑

k=1

αk+1〈〈(a+)kTn−1µ 1, 1〉〉.

For all k ≥ 1, (a+)kTn−1µ 1 ∈ Γ(H)ªC1, and thus, 〈〈(a+)kTn−1

µ 1, 1〉〉 = 0. Hence,we obtain:

µn = 〈〈a−Tn−1µ 1, 1〉〉+ α1µn−1

= 〈〈[a−, Tn−1µ ]1, 1〉〉+ 〈〈Tn−1

µ a−1, 1〉〉+ α1µn−1

=n−2∑

j=0

〈〈Tn−2−jµ [a−, Tµ]T j

µ1, 1〉〉+ α1µn−1.

We have:

[a−, Tµ] = [a−, a− + α1I +∞∑

k=1

αk+1(a+)k]

=∞∑

k=1

αk+1[a−, (a+)k]

=∞∑

k=1

αk+1

k−1∑r=0

(a+)k−1−r[a−, a+](a+)r

=∞∑

k=1

αk+1

k−1∑r=0

(a+)k−1−rP0(a+)r.

28 AUREL I. STAN

Now, we make the crucial observation that, for all r ≥ 1, P0(a+)r = 0, due to thefact that the range of (a+)r is H⊗r ⊕ H⊗(r+1) ⊗ · · · which is orthogonal to thevacuum space C1 (the range of P0). Thus in the sum

∑k−1r=0(a+)k−1−rP0(a+)r,

from the commutator [a−, Tµ], only the term corresponding to r = 0 survives.Therefore, we get:

[a−, Tµ] =∞∑

k=1

αk+1(a+)k−1P0.

It follows now, that:

µn =n−2∑

j=0

〈〈Tn−2−jµ [a−, Tµ]T j

µ1, 1〉〉+ α1µn−1

= α1µn−1 +n−2∑

j=0

∞∑

k=1

αk+1〈〈Tn−2−jµ (a+)k−1P0T

jµ1, 1〉〉.

Since

P0Tjµ1 = 〈〈T j

µ1, 1〉〉1= E[T j

µ]1= µj1,

we obtain:

µn = α1µn−1 +n−2∑

j=0

µj

∞∑

k=1

αk+1〈〈Tn−2−jµ (a+)k−11, 1〉〉

= α1µn−1 + α2

n−2∑

j=0

µj〈〈Tn−2−jµ 1, 1〉〉

+n−2∑

j=0

µj

∞∑

k=2

αk+1〈〈Tn−2−jµ (a+)k−11, 1〉〉

= α1µn−1 + α2

n−2∑

j=0

µjµn−2−j

+n−2∑

j=0

µj

∞∑

k=1

αk+2〈〈Tn−2−jµ (a+)k1, 1〉〉.

In the last sum:n−2∑

j=0

µj

∞∑

k=1

αk+2〈〈Tn−2−jµ (a+)k1, 1〉〉,

j is actually running from 0 to n− 3, since for j = n− 2, we have:

〈〈Tn−2−jµ (a+)k1, 1〉〉 = 〈〈(a+)k1, 1〉〉

= 〈〈(a+)k−11, a−1〉〉= 0,


for all k ≥ 1.We will now use the dual commutator method, to bring the creation operators

from right to left. In the last sum:

n−3∑

j=0

µj

∞∑

k=1

αk+2〈〈Tn−2−jµ (a+)k1, 1〉〉,

we swap Tn−2−jµ and (a+)k using the commutator formula:

[Tn−2−jµ , (a+)k]1 =

n−3−j∑

i=0

Tn−3−j−iµ [Tµ, (a+)k]T i

µ1

=n−3−j∑

i=0

Tn−3−j−iµ [a−, (a+)k]T i

µ1

=n−3−j∑

i=0

Tn−3−j−iµ (a+)k−1P0T

iµ1

=n−3−j∑

i=0

µiTn−3−j−iµ (a+)k−11.

Thus, since after the swap 〈〈(a+)kTn−2−jµ 1, 1〉〉 = 0, we obtain:

µn = α1µn−1 + α2

n−2∑

j=0

µjµn−2−j

+n−3∑

j=0

µj

∞∑

k=1

αk+2

n−3−j∑

i=0

µi〈〈Tn−3−j−iµ (a+)k−11, 1〉〉

= α1µn−1 + α2

n−2∑

j=0

µjµn−2−j

+n−3∑

j=0

µjα3

n−3−j∑

i=0

µi〈〈Tn−3−j−iµ 1, 1〉〉

+n−3∑

j=0

µj

∞∑

k=2

αk+2

n−3−j∑

i=0

µi〈〈Tn−3−j−iµ (a+)k−11, 1〉〉

= α1µn−1 + α2

n−2∑

j=0

µjµn−2−j

+ α3

n−3∑

j=0

n−3−j∑

i=0

µjµiµn−3−j−i

+n−3∑

j=0

µj

∞∑

k=1

αk+3

n−3−j∑

i=0

µi〈〈Tn−3−j−iµ (a+)k1, 1〉〉.

30 AUREL I. STAN

We observe, as before, that in the last sum:n−3∑

j=0

µj

∞∑

k=1

αk+3

n−3−j∑

i=0

µi〈〈Tn−3−j−iµ (a+)k1, 1〉〉,

j is actually running from 0 to n − 4, and i from 0 to n − 4 − j. We repeat thisprocedure swapping now Tn−3−j−i

µ and (a+)k, and so on, each time reducing therunning interval for j, until this interval disappears. It is now clear that in theend, we get:

µn = α1µn−1 + α2

∑

j1+j2=n−2

µj1µj2 + α3

∑

j1+j2+j3=n−3

µj1µj2µj3 + · · ·

+ αn

∑

j1+j2+···+jn=0

µj1µj2 · · ·µjn , (4.16)

for all n ≥ 1. Formula (4.16) is very interesting and easy to memorize.Dividing first both sides of formula (4.16) by wn+1, and then summing up from

n = 1 to ∞, we get:∞∑

n=1

µn

wn+1=

1w

α1

∞∑n=1

µn−1

wn

+1w

α2

[ ∞∑n=1

µn−1

wn

]2

+1w

α3

[ ∞∑n=1

µn−1

wn

]3

· · · · · · .

Since µ0 = 1, this means:

G(w)− 1w

=G(w)

w

∞∑

k=0

αk+1 [G(w)]k

=G(w)

wR(G(w)).

This is equivalent to:

wG(w) = G(w)R(G(w)) + 1,

which means:

w = R(G(w)) +1

G(w)= K(G(w)).

Thus G(w) and K(z) are inverses with respect to composition. ¤

Acknowledgement. The author would like to thank the referee for giving himmany important suggestions about how to improve this paper. Thus Claim 1,from Example 3.2, and Proposition 4.2 were added to the paper following his/herrecommendation.


References

1. Accardi, L., Kuo, H.-H., and Stan, A. I.: Characterization of probability measures throughthe canonically associated interacting Fock spaces, Infin. Dimens. Anal. Quantum Probab.Relat. Top. 7, No. 4 (2004) 485–505.

2. Accardi, L., Kuo, H.-H., and Stan, A. I.: Moments and commutators of probability measures,Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10, No. 4 (2007) 591–612.

3. Asai, N., Kubo, I., and Kuo, H.-H.: Multiplicative renormalization and generating functionsI, Taiwanese J. Math. 7 (2003) 89–101.

4. Asai, N., Kubo, I., and Kuo, H.-H.: Generating functions of orthogonal polynomials andSzego–Jacobi parameters, Prob. Math. Stat. 23 (2003) 273–291.

5. Asai, N., Kubo, I., and Kuo, H.-H.: Multiplicative renormalization and generating functionsII, Taiwanese J. Math. 8 (2004) 593–628.

6. Chihara, T. S.: An Introduction to Orthogonal Polynomials, Gordon & Breach, New York,1978.

7. Namli, S.: Multiplicative Renormalization Method for Orthogonal Polynomials, Ph.D. thesiselectonically available at http://etd.lsu.edu/docs/available/etd-11162007-115010.

8. Stan, A. I. and Whitaker, J. J.: A study of probability measures through commutators, . J.Theor. Prob., to appear.

9. Szego, M.: Orthogonal Polynomials. Coll. Publ. 23, Amer. Math. Soc., 1975.10. Voiculescu, D. V., Dykema, K. J., and Nica, A.: Free Random Variables, Vol. 1, CRM

Monograph Series, American Mathematical Society, Providence, Rhode Island USA, 1992.

Aurel I. Stan: Department of Mathematics, The Ohio State University at Marion,Marion, OH 43302, U.S.A.

E-mail address: [email protected]

A STOCHASTIC PROCESS ASSOCIATED WITH THEWEIGHTED WHITE NOISE DIFFERENTIATION

ISSEI KITAGAWA

Abstract. In this paper, we give a relationship between the weighted whitenoise differentiation and the Levy Laplacian introducing an operator changingthe white noise B(t) by B(t)2. In addition, we give an infinite dimensionalstochastic process generated by the weighted white noise differentiation anda relationship between the stochastic process and the Levy Laplacian.

1. Introduction

In [2], Kondratiev and Streit introduced a family (E)∗β , 0 ≤ β < 1, of generalizedfunctions and a family (E)β , 0 ≤ β < 1 of test functions. Let E = S(R) be theSchwartz space of rapidly decreasing functions on R. Then, taking a completeorthonormal basis ζn∞n=0 ⊂ E for L2(T ) with a fixed finite interval T ⊂ R, wedefine the generalized Levy Laplacian ∆L(h)Φ of Φ ∈ (E)∗β by

S[∆L(h)Φ](ξ) = limN→∞

1N

N−1∑n=0

S[Φ]′′(ξ)(hζn, hζn)

for each ξ in the complexification Ec of E and h ∈ E with supp(h) ⊂ T . Define anoperator L on (E)β by

Lϕ = S−1[Sϕ(ξ2)], ϕ ∈ (E)β .

Then the operator L is a continuous linear operator from (E)β into (E)∗β for12 ≤ β < 1. For 1

2 ≤ β < 1, we define an operator ∆L(h) on L[(E)β ] by∆L(h)Lϕ =

∑∞n=0 ∆L(h)Lϕn for ϕ =

∑∞n=0 ϕn ∈ (E)β , and denote ∆L(h) by

the same notation ∆L(h). Hence the operator ∆L(h) is a continuous linear oper-ator from L[(E)β ] into (E)∗β . Moreover the operator ∆L(h) is a continuous linearoperator from L[(E)β ] into itself. Let Dh =

∫T

h(u) ∂∂x(u)du, x ∈ E∗, for h ∈ E with

supp(h) ⊂ T . The weighted white noise differentiation Dh is a continuous linearoperator from (E)β into itself (see [4]). This operator on (E)β is same as the Grossdifferentiation on (E)β . Then we can give a relationship between the operator Dh

and the generalized Levy Laplacian ∆L(h) through the operator L as12∆L(h)Lϕ =

1|T |LDh2ϕ

2000 Mathematics Subject Classification. Primary 60H40.Key words and phrases. White noise theory, weighted white noise differentiation, Levy

Laplacian.

33



34 ISSEI KITAGAWA

for any ϕ ∈ (E)β . Hence we have also

e−t2 |T |∆L(h)Lϕ = L[e−tDh2 ϕ]

for t ≥ 0 and ϕ ∈ (E)β . Let ek∞k=0 be an orthonormal basis for L2(R). Definean infinite dimensional stochastic process X(t) by

X(t) = −tet∞∑

k=0

eiXk(t)〈h2, ek〉ek, t ≥ 0,

for h ∈ E with supp(h) ⊂ T , where Xk(t); t ≥ 0, k = 0, 1, 2, . . . , is a sequenceconsisting of independent Cauchy processes with the characteristic functions givenby

E[eizXk(t)

]= e−t|z|, z ∈ R, k = 0, 1, 2, . . . .

Then X(t) is an Ec-valued stochastic process generated by Dh2S ≡ SDh2 onS[(E)β ]. For y ∈ Ec let Ty be a translation operator defined on (E)β by

S(Tyϕ)(ξ) = Sϕ(ξ + y), ϕ ∈ (E)β .

Then we havee−

t2 |T |∆L(h)Lϕ = L[E[TX(t)ϕ]], t ≥ 0

for any ϕ ∈ (E)β .In this paper, we give a relationship between the weighted white noise dif-

ferentation and the Levy Laplacian [4, 6, 7, 8] acting on a space of white noisedistributions introducing an operator changing the white noise B(t) by B(t)2.Moreover based on infinitely many Cauchy processes, we give a relationship be-tween an infinite dimensional stochastic process generated by the weighted whitenoise differentiation and the Levy Laplacian.

The paper is organized as follows. In Section 2 we summarize some basic defi-nitions and results in the white noise analysis. In Section 3 we give a relationshipbetween the weighted white noise differentiation and the Levy Laplacian actingon a space of white noise distributions introducing an operator changing the whitenoise B(t) by B(t)2. In Section 4, based on infinitely many Cauchy processes, wegive an infinite dimensional stochastic process generated by the weighted whitenoise differentiation and a relationship between the stochastic process and theLevy Laplacian.

2. White Noise Background

Let L2(R) be a real separable Hilbert space with norm | · |0 and E = S(R)be the Schwartz space of rapidly decreasing functions on R. Let A = −d2/du2 +u2 + 1. This is a densely defined self-adjoint operator on L2(R) and there existsan orthonormal basis eν ; ν ≥ 0 for L2(R) such that Aeν = (2ν + 2)eν , ν =0, 1, 2, . . . . We define the norm | · |p by |f |p = |Apf |0 for f ∈ E and p ∈ R. LetEp = f ∈ E ; |f |p < ∞. Then Ep is a real separable Hilbert space with the norm| · |p and the dual space E ′p of Ep is the same as E−p. Let E be the projective limitspace of Ep; p ≥ 0 and E ′ the dual space of E . Then E becomes a nuclear spacewith the Gel’fand triple E ⊂ L2(R) ⊂ E ′ . We denote the complexifications ofL2(R), E and Ep by L2

c(R), Ec and Ec,p, respectively.

WEIGHTED WHITE NOISE DIFFERENTIATION AND LEVY LAPLACIAN 35

By the Bochner-Minlos theorem there is a unique probability measure µ on E ′such that ∫

E′expi〈x, ξ〉dµ(x) = exp

(−1

2|ξ|20

), ξ ∈ E ,

where 〈·, ·〉 is the canonical bilinear form on E ′ × E .The space (L2) = L2(E ′ , µ) of complex-valued square-integrable functionals

defined on E ′ admits the well-known Wiener-Ito decomposition :

(L2) =∞⊕

n=0

Hn,

where Hn is the space of multiple Wiener integrals of order n ∈ N and H0 = c.Let L2

c(R)b⊗n denote the n-fold symmetric tensor product of L2c(R). If ϕ ∈ (L2)

is represented by ϕ =∑∞

n=0 In(fn), fn ∈ L2c(R)b⊗n, then the (L2)-norm ‖ϕ‖0 is

given by

‖ϕ‖0 =

( ∞∑n=0

n!|fn|20) 1

2

,

where | · |0 means also the norm of L2c(R)b⊗n.

For p ∈ R, let ‖ϕ‖p = ‖Γ(A)pϕ‖0, where Γ(A) is the second quantizationoperator of A. If p ≥ 0, let (Ep) be the domain of Γ(A)p. If p < 0, let (Ep) be thecompletion of (L2) with respect to the norm ‖ · ‖p. Then (Ep), p ∈ R, is a Hilbertspace with the norm ‖ · ‖p. It is easy to see that for p > 0, the dual space (Ep)∗ of(Ep) is given by (E−p). Moreover, for any p ∈ R, we have the decomposition

(Ep) =∞⊕

n=0

H(p)n ,

where H(p)n is the completion of In(f); f ∈ E b⊗n

c with respect to ‖·‖p. Here E b⊗nc is

the n-fold symmetric tensor product of Ec. We also have H(p)n = In(f); f ∈ E b⊗n

c,p for any p ∈ R, where E b⊗n

c,p is also the n-fold symmetric tensor product of Ec,p. Thenorm ‖ϕ‖p of ϕ =

∑∞n=0 In(fn) ∈ (Ep) is given by

‖ϕ‖p =

( ∞∑n=0

n!|fn|2p) 1

2

, fn ∈ E b⊗nc,p ,

where the norm of E b⊗nc,p is denoted also by | · |p.

Let 0 ≤ β < 1 be a fixed number. We define the norm ‖ · ‖p,β by

‖ϕ‖p,β =

( ∞∑n=0

(n!)1+β |fn|2p) 1

2

, fn ∈ E b⊗nc,p .

Let (Ep)β = ϕ ∈ (L2); ‖ϕ‖p,β < ∞. Then (Ep)β is a separable Hilbert space withthe norm ‖·‖p,β and the dual space (Ep)∗β of (Ep)β is the same as (E−p)−β . Let (E)β

be the projective limit space of (Ep)β ; p ≥ 0 and (E)∗β the dual space of (E)β .

36 ISSEI KITAGAWA

Then (E)β becomes a nuclear space with the Gel’fand triple (E)β ⊂ (L2) ⊂ (E)∗β .We denote by ¿ ·, · À the canonical bilinear form on (E)∗β × (E)β . Then we have

¿ Φ, ϕ À=∞∑

n=0

n!〈Fn, fn〉

for any Φ =∑∞

n=0 In(Fn) ∈ (E)∗β and ϕ =∑∞

n=0 In(fn) ∈ (E)β , where the canon-ical bilinear form on (E⊗n

c )∗ × (E⊗nc ) is denoted also by 〈·, ·〉.

Since exp〈·, ξ〉 ∈ (E)β , the S-transform is defined on (E)∗β by

S[Φ](ξ) = exp(−1

2〈ξ, ξ〉

)¿ Φ, exp〈·, ξ〉 À, ξ ∈ Ec.

We have the following characterization theorem of the S-transform.

Theorem 2.1 ([2, 4]). A complex-valued function F on Ec is the S-transform ofan element in (E)∗β if and only if F satisfies the conditions :

1) For any ξ and η in Ec, the function F (zξ + η) is an entire function ofz ∈ C.

2) There exist nonegative constants K, a, and p such that

|F (ξ)| ≤ K exp[a|ξ|

21−βp

], ∀ξ ∈ Ec.

Consider F = S[Φ] with Φ ∈ (E)∗β . By Theorem 2.1, for any ξ, η ∈ Ec thefunction F (ξ + zη) admits the series expansion:

F (ξ + zη) =∞∑

n=0

zn

n!F (n)(ξ)(η, . . . , η),

where F (n)(ξ) : Ec × · · · × Ec → C is a continuous n-linear functional.Fix a finite interval T in R. Take an orthonormal basis ζn∞n=0 ⊂ E for L2(T )

satisfying the equally dense and uniform boundedness property. Let DL denotethe set of all Φ ∈ (E)∗β such that the limit

∆L(h)S[Φ](ξ) = limN→∞

1N

N−1∑n=0

S[Φ]′′(ξ)(hζn, hζn)

exists for any ξ ∈ Ec and h ∈ E with supp(h) ⊂ T . The generalized Levy Laplacian∆L(h) is defined by

∆L(h)Φ = S−1∆L(h)SΦ, Φ ∈ DL.

3. Weighted White Noise Differentiation and the Levy Laplacian

Define an operator L on (E)β by

Lϕ = S−1[Sϕ(ξ2)], ϕ ∈ (E)β .

Then we have the following theorem.

Theorem 3.1. Let 12 ≤ β < 1. For all ϕ ∈ (E)β, Lϕ is in (E)∗β, the operator L is

a continuous linear operator from (E)β into (E)∗β.


Proof. For ϕ =∑∞

n=0 In(fn) ∈ (E)β , we have

Lϕ =∞∑

n=0

∫

Rn

fn(u1, . . . , un) : x(u1)2 · · ·x(un)2 : du

=∞∑

n=0

∫

R2n

∫

Rn

fn(u1, . . . , un)δ⊗2u1⊗ · · · ⊗δ⊗2

un(v1, . . . , v2n)du

: x(v1) · · ·x(v2n) : dv1 · · · dv2n.

Denote Gn(v1, . . . , v2n) =∫Rn fn(u1, . . . , un)δ⊗2

u1⊗ · · · ⊗δ⊗2

un(v1, . . . , v2n)du

and gk1,··· ,k2n(u1, · · · , un) = ek1(u1) · · · ek2n

(un). Then we obtain

〈Gn, ek1 ⊗ · · · ⊗ ek2n〉2

=(∫

Rn

fn(u1, . . . , un)ek1(u1)ek2(u1) · · · ek2n(un)du

)2

≤ |fn|2q|gk1,··· ,k2n |2−q

= |fn|2q∞∑

l1,··· ,l2n=0

(2l1 + 2)−2q · · · (2l2n + 2)−2q|〈gk1,··· ,k2n , el1 ⊗ · · · ⊗ el2n〉2|

= |fn|2q(2k1 + 2)−2q · · · (2k2n + 2)−2q

for some q > 0. Hence for any p > 0 and 12 ≤ β < 1 there exists q > 0 such that

‖Lϕ‖2−p,−β =∞∑

n=0

(2n)!1−β |Gn|2−p,−β

=∞∑

n=0

(2n)!1−β∞∑

k1,··· ,k2n=0

(2k1 + 2)−2p · · · (2k2n + 2)−2p×

|〈Gn, ek1 ⊗ · · · ⊗ ek2n〉2|

≤∞∑

n=0

(2n)!1−β∞∑

k1,··· ,k2n=0

(2k1 + 2)−2(p+q) · · · (2k2n + 2)−2(p+q)|fn|2q

=∞∑

n=0

(n!)1+β (2n)!1−β

(n!)1+β

( ∞∑

k=0

(2k + 2)−2(p+q)

)2n

|fn|2q.

For β ≥ 12 , there exists N ∈ N such that (2n)!1−β < (n!)1+β for all n ≥ N . Hence

for β ≥ 12 we obtain

‖Lϕ‖2−p,−β ≤N−1∑n=0

(n!)1+β (2n)!1−β

(n!)1+β

( ∞∑

k=0

(2k + 2)−2(p+q)

)2n

|fn|2q

+∞∑

n=N

(n!)1+β

( ∞∑

k=0

(2k + 2)−2(p+q)

)2n

|fn|2q.

38 ISSEI KITAGAWA

Define M = max

(2n)!1−β

(n!)1+β | n = 0, 1, 2, . . . , N − 1

. Then for large p > 0 thereexists q > 0 such that

‖Lϕ‖2−p,−β ≤ M

∞∑n=0

(n!)1+β

( ∞∑

k=0

(2k + 2)−2(p+q)

)2n

|fn|2q

≤ M

∞∑n=0

(n!)1+β |fn|2q

= M‖ϕ‖2q,β .

The poof is completed. ¤

We define an operator ∆L(h) on L[(E)β ] by

∆L(h)Lϕ =∞∑

n=0

∆L(h)Lϕn for ϕ =∞∑

n=0

ϕn ∈ (E)β ,

and denote ∆L(h) also by the same notation ∆L(h).

Theorem 3.2. Let 12 ≤ β < 1. The operator ∆L(h) is a continuous linear operator

from L[(E)β ] into (E)∗β.

Proof. For ϕ =∑∞

n=0 In(fn) ∈ (E)β , we have∞∑

n=0

∆L(h)L[In(fn)]

=∞∑

n=0

2n

∫

Rn

h(un)2fn(u1, . . . , un) : x(u1)2 · · ·x(un−1)2 : du

=∞∑

n=0

2n

∫

R2(n−1)

∫

Rn

h(un)2fn(u1, . . . , un)δ⊗2u1⊗ · · · ⊗δ⊗2

un−1(v1, . . . , v2(n−1))du

: x(v1) · · ·x(v2(n−1)) : dv1 · · · dv2(n−1).

Let Hn(v1, . . . , v2(n−1)) = 2n∫Rn h(un)2fn(u)δ⊗2

u1⊗ · · · ⊗δ⊗2

un−1(v1, . . . , v2(n−1))du.

Then for p > 0 and 12 ≤ β < 1, it holds that

∞∑n=0

(2n− 2)!1−β |Hn|2−p,−β =∞∑

n=0

(2n− 2)!1−β∞∑

k1,··· ,k2(n−1)=0

(2k1 + 2)−2p · · ·

(2k2(n−1) + 2)−2p|〈Hn, ek1 ⊗ · · · ⊗ ek2(n−1)〉2|.Let gk1,··· ,k2(n−1)(u1, · · · , un−1) = ek1(u1) · · · ek2(n−1)(un−1). Then we obtain

〈Hn, ek1 ⊗ · · · ⊗ ek2(n−1)〉2

=(

2n

∫

Rn

h(un)2fn(u1, . . . , un)ek1(u1)ek2(u1) · · · ek2(n−1)(un−1)du)2

≤ (2n)2|h|4∞|fn|2q|gk1,··· ,k2(n−1) |2−q


for some q > 0. Since

|gk1,··· ,k2(n−1) |2−q

=∞∑

l1,··· ,l2(n−1)=0

(2l1 + 2)−2q · · · (2l2(n−1) + 2)−2q×

|〈gk1,··· ,k2(n−1) , el1 ⊗ · · · ⊗ el2(n−1)〉2|

=∞∑

l1,··· ,l2(n−1)=0

(2l1 + 2)−2q · · · (2l2(n−1) + 2)−2qδk1,l1 · · · δk2(n−1),l2(n−1)

= (2k1 + 2)−2q · · · (2k2(n−1) + 2)−2q,

we see that for any p > 0 there exists q > 0 such that

‖∆L(h)Lϕ‖2−p,−β

≤ |h|4∞∞∑

n=0

(2n− 2)!1−β(2n)2( ∞∑

k=0

(2k + 2)−2(p+q)

)2(n−1)

|fn|2q

= |h|4∞∞∑

n=0

(n!)1+β (2n− 2)!1−β(2n)2

(n!)1+β

( ∞∑

k=0

(2k + 2)−2(p+q)

)2(n−1)

|fn|2q.

For β ≥ 12 , there exists N ∈ N such that (2n − 2)!1−β(2n)2 ≤ (n!)1+β for all

n ≥ N . Hence for β ≥ 12 we obtain

‖∆L(h)Lϕ‖2−p,−β

≤ |h|4∞N−1∑n=0

(n!)1+β (2n− 2)!1−β(2n)2

(n!)1+β

( ∞∑

k=0

(2k + 2)−2(p+q)

)2(n−1)

|fn|2q

+ |h|4∞∞∑

n=N

(n!)1+β

( ∞∑

k=0

(2k + 2)−2(p+q)

)2(n−1)

|fn|2q.

Define K = |h|4∞max

(2n−2)!1−β(2n)2

(n!)1+β | n = 0, 1, 2, . . . , N − 1

. Then for largep > 0 there exists q > 0 such that

‖∆L(h)Lϕ‖2−p,−β ≤ K

∞∑n=0

(n!)1+β

( ∞∑

k=0

(2k + 2)−2(p+q)

)2(n−1)

|fn|2q

≤ K

∞∑n=0

(n!)1+β |fn|2q

= K‖ϕ‖2q,β .

Thus by Theorem 3.1, the poof is completed. ¤

We can easily check that ∆L(h)Φ ∈ L[(E)β ] for each Φ ∈ L[(E)β ]. Then wehave the following corollary.

40 ISSEI KITAGAWA

Corollary 3.3. The operator ∆L(h) is a continuous linear operator from L[(E)β ]into itself.

Let Dh =∫

Th(u) ∂

∂x(u)du, x ∈ E∗, for h ∈ E with supp(h) ⊂ T . Then we havethe next theorem.

Theorem 3.4 ([4]). The weighted white noise differentiation Dh is a continuouslinear operator from (E)β into itself.

Example 3.5. For ϕ = : exp(c∫R

g(u)x(u)du) : , the equality holds :

∆L(h)ϕ =∞∑

n=0

cn

n!∆L(h) :

(∫

R

g(u)x(u)du

)n

: .

Theorem 3.6. For any ϕ ∈ (E)β, the equality holds :

12∆L(h)Lϕ =

1|T |LDh2ϕ. (3.1)

Proof. For ϕ = : exp(c∫R

g(u)x(u)du) : ∈ (E)β , g ∈ E , c ∈ C, we have

LDh2ϕ = c

∫

T

g(u)h(u)2duLϕ.

On the other hand, by Example 3.5, we have12∆L(h)Lϕ =

c

|T |∫

T

g(u)h(u)2duLϕ.

Since the linear span of : exp(c∫R

g(u)x(u)du) :, g ∈ E , c ∈ C is dense in (E)β , bythe continuities of L,Dh2 and ∆L(h) from Theorem 3.1, 3.4 and Corollary 3.3, weobtain (3.1). ¤

Corollary 3.7. For any t ≥ 0 and ϕ ∈ (E)β, the equality holds :

e−t2 |T |∆L(h)Lϕ = L[e−tDh2 ϕ]. (3.2)


g(u)x(u)du) : ∈ (E)β , g ∈ E , c ∈ C. By Theorem 3.1,3.6, we have

e−t2 |T |∆L(h)Lϕ =

∞∑n=0

(− t2 |T |)n

n!∆n

L(h)Lϕ

=∞∑

n=0

(−t)ncn

n!

(∫

T

g(u)h(u)2du

)n

Lϕ

= L

[ ∞∑n=0

(−t)ncn

n!

(∫

T

g(u)h(u)2du

)n

ϕ

]

= L[e−tDh2 ϕ].


g(u)x(u)du) :, g ∈ E , c ∈ C is dense in (E)β , bythe continuities of L,Dh2 and ∆L(h) from Theorem 3.1, 3.4 and Corollary 3.3, weobtain (3.2). ¤


4. An Infinite Dimensional Stochastic Process

Let Xk(t); t ≥ 0, k = 0, 1, 2, . . . , be an infinite sequence consisting of inde-pendent Cauchy processes with the characteristic functions are given by

E[eizXk(t)

]= e−t|z|, z ∈ R, k = 0, 1, 2, . . . .

Let X(t); t ≥ 0 be an infinite dimensional stochastic process defined by

X(t) = −tet∞∑

k=0

eiXk(t)〈h2, ek〉ek, t ≥ 0,

for h ∈ E with supp(h) ⊂ T .

Proposition 4.1. For all t ≥ 0 we have X(t) ∈ Ec (a.e.).

Proof. For all p ∈ R, we have

E[|X(t)|2p

]= E

[ ∞∑ν=0

(2ν + 2)2p|〈X(t), eν〉|2]

=∞∑

ν=0

(2ν + 2)2pE[|〈X(t), eν〉|2

].

Since

E[|〈X(t), eν〉|2

]= E

∣∣∣∣∣tet

⟨ ∞∑

k=0

eiXk(t)〈h2, ek〉ek, eν

⟩∣∣∣∣∣

2

= E

[t2e2t

∣∣∣eiXν(t)〈h2, eν〉∣∣∣2]

= t2e2tE

[∣∣∣eiXν(t)∣∣∣2 ∣∣〈h2, eν〉

∣∣2]

= t2e2t∣∣〈h2, eν〉

∣∣2 ,

we obtain

E[|X(t)|2p

]= t2e2t

∞∑ν=0

(2ν + 2)2p∣∣〈h2, eν〉

∣∣2

= t2e2t|h2|2p.Hence for all p ∈ R, E

[|X(t)|2p]

< ∞ holds. Therefore we have X(t) ∈ Ec

(a.e.). ¤

For y ∈ Ec let Ty be a translation operator defined on (E)β by

S(Tyϕ)(ξ) = Sϕ(ξ + y), ϕ ∈ (E)β .

Theorem 4.2 ([4]). Let y ∈ E ′c. For any p ≥ 0, q > 0 with |y|−p < ∞ and22q−1+β ≥ 1, it holds that

‖Tyϕ‖p,β ≤ ‖ϕ‖p+q,β(1− 2−2q+1−β)−12 exp

[(1 + β)4−

q+β1+β |y|

21+β

−p

], ϕ ∈ (E)β .

42 ISSEI KITAGAWA

Lemma 4.3. Let E be a linear span of : exp(c∫R

g(u)x(u)du) :, g ∈ E, c ∈ C.Then, for any ϕ ∈ (E)β, there exists a sequence (ϕl)∞l=1 ⊂ E such that

E[STX(t)ϕ(ξ)] = liml→∞

E[STX(t)ϕl(ξ)].

Proof. Since X(t) ∈ Ec, by Theorem 4.2 there exists K > 0 such that

E[‖TX(t)ϕ‖2p,β

] ≤ K‖ϕ‖2p+q,β , ϕ ∈ (E)β ,

for any p ≥ 0 and q > 0. Since E is dense in (E)β (see [1]), for any ϕ ∈ (E)β thereexists a sequence (ϕl)∞l=1 ⊂ E such that ‖ϕl−ϕ‖p,β → 0 as l →∞, for all p. Hencewe have

E[|STX(t)ϕl(ξ)− STX(t)ϕ(ξ)|] = E[|〈TX(t)(ϕl − ϕ), φξ〉|]≤ E[‖TX(t)(ϕl − ϕ)‖p,β ]‖φξ‖−p,−β

≤ K‖ϕl − ϕ‖p+q,β‖φξ‖−p,−β .

Consequently we obtain

E[|STX(t)ϕl(ξ)− STX(t)ϕ(ξ)|] ≤ K‖ϕl − ϕ‖p+q,β‖φξ‖−p,−β

→ 0 (l →∞).

¤

Theorem 4.4. The stochastic process X(t) is generated by Dh2S ≡ SDh2 .


g(u)x(u)du) : ∈ (E)β , g ∈ E , c ∈ C, we have

E[Sϕ(ξ + X(t))]

=∞∑

n=0

cn

n!E

[∫

R

g(u)ξ(u)du− tet

∫

T

g(u)∞∑

k=0

eiXk(t)〈h2, ek〉ek(u)du

n]

=∞∑

n=0

cn

n!

(∫

R

g(u)ξ(u)du− t

∫

T

g(u)h(u)2du

)n

= exp(−ct

∫

T

g(u)h(u)2du

)Sϕ(ξ)

= e−tgDh2 Sϕ(ξ).


g(u)x(u)du) :, g ∈ E , c ∈ C is dense in (E)β ,and by the continuity of e−tgDh2 and Lemma 4.3, the proof is completed. ¤

Theorem 4.5. For any ϕ ∈ (E)β, the following equality holds

e−t2 |T |∆L(h)Lϕ = L[E[TX(t)ϕ]], t ≥ 0. (4.1)

Proof. By Proposition 4.1 for ϕ ∈ (E)β , we have

S(TX(t)ϕ)(ξ2) = Sϕ(ξ2 + X(t)).


Hence for ϕ = : exp(c∫R

g(u)x(u)du) : ∈ (E)β , g ∈ E , c ∈ C, we see that

E[S(TX(t)ϕ)(ξ2)]

=∞∑

n=0

cn

n!E

[(∫

R

g(u)ξ(u)2du− tet

∫

T

g(u)∞∑

k=0

eiXk(t)〈h2, ek〉ek(u)du

)n]

=∞∑

n=0

cn

n!

(∫

R

g(u)ξ(u)2du− t

∫

T

g(u)h(u)2du

)n

= exp(−ct

∫

T

g(u)h(u)2du

)Sϕ(ξ2).

On the other hand, by Corollary 3.7 for ϕ = : exp(c∫R

g(u)x(u)du) : ∈ (E)β ,g ∈ E , c ∈ C, we have

e−t2 |T |e∆L(h)SLϕ(ξ) = exp

(−ct

∫

T

g(u)h(u)2du

)Sϕ(ξ2).


g(u)x(u)du) :, g ∈ E , c ∈ C is dense in (E)β , byCorollary 3.7 and Lemma 4.3, we obtain (4.1) for all ϕ ∈ (E)β . ¤

Acknowledgement. The author would like to express his deep appreciation toProfessors T.Hida, H-H. Kuo, and K. Saito for their advice.

References

1. Hida, T.: Brownian Motion, Springer-Verlag, 1980.2. Kondratiev, Yu. G. and Streit, L.: Spaces of white noise distributions: Constructions, De-

scriptions, Applications. I, Reports on Math. Phys. 33 (1993) 341–366.3. Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise I-IV, Proc. Japan Acad. 56A

(1980) 376–380, 56A (1980) 411–416, 57A (1981) 433–436, 58A (1982) 186–189.4. Kuo, H.-H.: White Noise Distribution Theory, CRC Press, 1996.5. Kuo, H.-H.: Introduction to Stochastic Integration, Universitext (UTX), Springer, 2006.6. Kuo, H.-H., Obata, N. and Saito, K.: Diagonalization of the Levy Laplacian and related

stable processes, Infinite Dimensional Analysis, Quantum Probab. and Related Topics. 5(2002) 317–331.

7. Levy, P.: Lecons d’Analyse Fonctionnelle, Gauthier-Villars, Paris, 1922.8. Saito, K.: The Levy Laplacian and stable processes, in: Choas, Soliton and Fractals 12

(2001) 2865–2872.

Issei Kitagawa: Department of Mathematics, Meijo University, Nagoya 468-8502,Japan


STOCHASTIC HEAT EQUATION WITH INFINITEDIMENSIONAL FRACTIONAL NOISE: L2-THEORY

RALUCA BALAN

Abstract. In this article we consider the stochastic heat equation in [0, T ]×Rd, driven by a sequence (βk)k of i.i.d. fractional Brownian motions of in-dex H > 1/2 and random multiplication functions (gk)k. The stochasticintegrals are of Hitsuda-Skorohod type and the solution is interpreted in theweak sense. Using Malliavin calculus techniques, we prove the existence anduniqueness of the solution in a certain space of random processes. Our resultis similar to the one obtained in [18] for the stochastic heat equation drivenby a sequence (wk)k of i.i.d. Brownian motions, in which case the stochasticintegrals are interpreted in the Ito sense.

1. Introduction

There is now a very rich theory dedicated to the study of stochastic partialdifferential equations (s.p.d.e.), which has been developed continuously during thepast three decades, one of its goals being to offer a solid mathematical explanationfor phenomena which evolve over time and are influenced by randomness.

Traditionally, the temporal structure of the noise perturbing such an equationwas that of a Brownian motion. In the recent years, there has been an increasedinterest in looking into the possibility of replacing this structure with that of afractional Brownian motion (fBm), which allows to build more flexibility into thetime component of the noise, (depending on the value of the Hurst parameter Hof the fBm), and increases the potential for applications.

The aim of the present article is to analyze the stochastic heat equation, in aspace of square-integrable functions, when the driving noise bears the structure ofthe fBm with index H > 1/2. Such a theory exists for equations whose noise termsbehave like the Brownian motion, in spaces of arbitrary summability exponentp ≥ 2 (the fundamental contribution is [17]; see [18] for more details), but hasnever been developed in the fBm case.

We recall that the fBm with Hurst parameter H ∈ (0, 1) is a zero-mean Gaussianprocess β = (βt)t∈[0,T ], with covariance

E(βtβs) = RH(t, s) :=12(t2H + s2H − |t− s|2H).

(The Brownian motion is a fBm with parameter H = 1/2.) The major difficulty isthe fact that the fBm is not a semimartingale, and hence one can not use the Ito

2000 Mathematics Subject Classification. Primary 60H15; Secondary 60H07.Key words and phrases. Fractional Brownian motion; Malliavin calculus; stochastic heat

equation.

45



46 RALUCA BALAN

integral and its associated stochastic calculus. (We refer the reader to [30], [15] formore details on the fBm, and to [35] for a careful analysis of various integrationquestions related to the fBm.)

To the best of our knowledge, three methods have been developed to circum-vent this difficulty, and they all perform well, especially in the regular case whenH > 1/2, which is the one considered in the present paper. One method usesthe Hitsuda-Skorohod integral and the associated Malliavin calculus (see [9], [10],[1], [2], [5] for some of the original developments). This method is used in thepresent article. Another method exploits the temporal “smoothness” of the noise,as opposed to the “roughness” of the Brownian path, and considers a pathwisegeneralized Stieltjes integral and its associated fractional calculus, as a replace-ment for the Ito-Stratonivich integral: the recent contributions [34], [37] show thefull power of this method at work; background reading on fractional integrationare [36], [40]. A third method relies on a re-thinking of the concept of noise, andthe analysis of the infinite-dimensional “rough paths”, as it was originally devel-oped in [25], [26]. So far, this last method has been used mostly for the studyof stochastic differential equations, the recent article [13] being among the firstattempts to analyze s.p.d.e.’s driven by rough paths.

On the other hand, there are many different approaches to the theory of s.p.d.e.’sin the literature, speaking only of equations whose noise term bears the temporalstructure of the Brownian motion, i.e. it is “white in time”. These approacheshave been developed in parallel, each of them being fruitful in its own way. Asignificant amount of effort is dedicated to unify to some extent these approaches.

To explain the contribution of the present paper, we need to recall briefly thesalient features of these approaches, without aiming at exhausting the whole listof references in this active area of research. The vehicle that we choose for thisoverview of the literature is the stochastic heat equation on [0, T ] × Rd, which isthe focus of investigation in the present article.

In the stochastic approach (also called the “Walsh approach”, due to [39]), whenH = 1/2, this equation is written as:

∂u

∂t= ∆u+ f(t, x) + h(t, x)W (dt, dx), t ∈ [0, T ], x ∈ Rd. (1.1)

where f, h are predictable functions, the noise W is defined as a collection Wt(ϕ)of Gaussian random variables with E(Wt(ϕ)Ws(ψ)) = (t ∧ s)〈ϕ,ψ〉U , and U is aHilbert space of functions on Rd. Initially, it was assumed that U = L2(Rd). Later,U became a particular Hilbert space of distributions in Rd, which is introduced viaa kernel measure Γ (see [28], [29], [6], [7], [33], [3]). The noise W is said to be whitein time, and “colored” in space. In short, the space U gives the color in space. TheWalsh approach has been extended to cover the fBm case, by replacing W witha “fractional-colored” noise B with covariance E(Bt(ϕ)Bs(ψ)) = RH(t, s)〈ϕ,ψ〉U(see [32], [23], [4]).

A broader color-spectrum in space can be achieved via the semigroup approach,treated comprehensively in [8], in the case H = 1/2 (see also [16], [22], [24]). Whenusing this approach, one works with an arbitrary Hilbert space U , and an infinite-dimensional Brownian motion (W t)t∈[0,T ], whose covariance is a nuclear operator

HEAT EQUATION WITH FRACTIONAL NOISE 47

Q. In this formulation, the equation is written as:

dXt = [∆Xt + F (t)]dt+H(t)dW t, t ∈ [0, T ]. (1.2)

Generally speaking, F,H are predictable functions with values in a suitable Hilbertspace V of functions on Rd, respectively in the space L2(U, V ) of Hilbert-Schmidtoperators from U to V . (If U is the particular Hilbert space of distributions on Rd

considered in the Walsh approach, corresponding to a measure Γ, and W is a cylin-drical Brownian motion on U , then Wt(ϕ) := 〈W t, ϕ〉U is a noise which is coloredin space, and (1.1) can be rewritten in form (1.2) with Xt = u(t, ·), F (t) = f(t, ·),H(t)ek = h(t, ·)ek and ek(x) =

∫Rd ek(x − y)Γ(dy), provided that ek 7→ h(t, ·)ek

lies in L2(U, V ).) The semigroup approach has been successfully used in the fBmcontext, by considering an infinite-dimensional fBm (Bt)t∈[0,T ]. Stochastic evolu-tion equations with this type of noise have been analyzed in [38], [27], using theHitsuda-Skorohod integral method, respectively the pathwise integral method; seealso [14], [11] for some earlier developments.

Finally, in the Lp-theory approach, the equation is written as: (see [18])

du = [∆u+ f(t, x)]dt+∞∑

k=1

gk(t, x)dwkt , t ∈ [0, T ]. (1.3)

Here f and g = (gk)k are predictable functions with values in the Sobolev spaceHn−1

p (Rd), respectively Hn+1p (Rd, l2), and the solution u lies in a subspace of

Lp(Ω × [0, T ];Hnp (Rd)), speaking only of the main properties (n is not necessar-

ily an integer). One of the appealing features of this approach is the relativelysimple structure of the noise, given by a sequence (wk)k of i.i.d. Brownian mo-tions. However, a closer look reveals that in fact the multiplication functions gk

incorporate some of the spatial color of the noise, through the orthonormal ba-sis (ek)k of the space U (in the Walsh formulation). More precisely, by takingWt(ϕ) =

∑k w

kt 〈ϕ, ek〉U , we obtain a noise which is colored in space, and (1.1)

can be rewritten in form (1.3), with gk(t, x) = h(t, x)ek(x) (see [12]).As mentioned earlier, this last approach has not been considered yet in the

context of the fBm. The present paper is the first attempt to fill this gap, byusing the Hitsuda-Skorohod integral and the Malliavin calculus, as a replacementfor the Ito integral.

This article is organized as follows. In Section 2, we give the background ma-terial on the Malliavin calculus with respect to the fBm.

In Section 3, we develop some special Malliavin calculus techniques, suitablefor treating Hn

2 -valued random variables. A fundamental property which is usedin the present paper is that for an arbitrary function g ∈ D1,2

β (|HHn2|), the action

of the Gross-Malliavin derivative Dβg on a test function φ ∈ C∞0 , coincides withthe Gross-Malliavin derivative of the action of g on φ.

Section 4 contains a generalization of the second-moment maximal inequality forthe Hitsuda-Skorohod integral with respect to the fBm (due to [2]), to the case of aninfinite sequence of integrals with respect to some i.i.d. fBm’s. This inequality is ofcrucial importance in the present article, being the replacement of the Burkholder-Davis-Gundy inequality, which is used in the Ito calculus. Although the result of

48 RALUCA BALAN

[2] (in the case of a single fBm) has been proved for an arbitrary moment of orderp > 1/H, its generalization to the case of a sequence of fBm’s becomes morecomplex. The inequality of [2], which lies at the origin of our developments, hasbeen obtained only in the case H > 1/2. As far as we know, a similar inequalitydoes not exist for the case H < 1/2. This is the reason the case H < 1/2 is nottreated in the present article.

In Section 5, we introduce the concept of solution and we examine the solutionspace Hn

2,H . Our definition of the solution space can be compared with Defini-tion 3.1, [18], which introduces the solution space for a very general second-orders.p.d.e.’s of parabolic type (in particular the stochastic heat equation), whose noiseis given by a sequence of Brownian motions. However, there are two essential dif-ferences between these two definitions. One comes from the fact that in the fBm’scase, the coefficients f and gk are jointly measurable in (ω, t), but not necessarilypredictable; this can be viewed as a relaxation. The trade-off is that the coefficientsgk multiplying the fractional noise has an additional “differentiability” property inω (rigorously defined via the Malliavin calculus techniques), which is not neededin the case of the Brownian noise.

Section 6 contains the result about the existence and uniqueness of the solutionto the stochastic heat equation. The proof of this result is based on some prelim-inary estimates of the difference between the solution of the stochastic equationand the solution of the “deterministic” equation.

2. Malliavin Calculus Preliminaries

In this section we introduce the basic facts of Malliavin calculus with respectto the fBm of index H > 1/2. We refer the reader to [30], [31], [20], [21].

We begin by introducing various Hilbert spaces of deterministic functions, whichare used in the present article.

If V is an arbitrary Hilbert space, let EV be the class of all elementary functionsφ : [0, T ] → V of the form φ(t) =

∑mi=1 1(ti,ti+1](t)vi with 0 ≤ t1 < . . . < tm ≤ T

and vi ∈ V , and HV be the completion of EV with respect to the inner product〈·, ·〉HV

defined by:

〈φ, ψ〉HV := αH

∫ T

0

∫ T

0

〈φ(t), ψ(s)〉V |t− s|2H−2dsdt, αH = H(2H − 1).

Let |HV | be the space of all strongly measurable functions φ : [0, T ] → V with‖φ‖|HV | <∞, where

‖φ‖2|HV | := αH

∫ T

0

∫ T

0

‖φ(t)‖V ‖φ(s)‖V |t− s|2H−2dtds.

Then EV is dense in |HV | with respect to the norm ‖ · ‖|HV |. We have:

‖φ‖HV≤ ‖φ‖|HV | ≤ bH‖φ‖L1/H([0,T ];V ) ≤ bH‖φ‖L2([0,T ];V ), (2.1)

for a constant bH > 0, and L2([0, T ];V ) ⊂ L1/H([0, T ];V ) ⊂ |HV | ⊂ HV . IfV = R, we let EV = E , HV = H and |HV | = |H|. Note that HV ' H⊗ V .


We denote by H ⊗ |HV | the space of all strongly measurable functions φ :[0, T ]2 → V with ‖φ‖H⊗|HV | <∞, where

‖φ‖2H⊗|HV | := α2H

∫

[0,T ]4‖φ(t, θ)‖V ‖φ(s, η)‖V |t− s|2H−2 |θ − η|2H−2dθdηdsdt.

In particular,

〈φ, ψ〉H⊗H := α2H

∫

[0,T ]4φ(t, θ)ψ(s, η)|t− s|2H−2||θ − η|2H−2dθdηdsdt,

and we hav:e L2([0, T ]2) ⊂ L1/H([0, T ]2) ⊂ H⊗ |H| ⊂ H ⊗H, with

‖φ‖H⊗H ≤ ‖φ‖H⊗|H| ≤ bH‖φ‖L1/H([0,T ]2) ≤ bH‖φ‖L2([0,T ]2). (2.2)

We are now ready to introduce the main ingredients of the Malliavin calculus.Let β = (βt)t∈[0,T ] be a fBm of Hurst index H > 1/2.

One can see that H is the completion of E with respect to the inner product

〈1[0,t], 1[0,s]〉H = RH(t, s).

The map t 7→ βt can be extended to an isometry between H and the Gaussianspace associated with β. We denote this isometry by φ 7→ β(φ).

Let Sβ := F = f(β(φ1), . . . , β(φn)); f ∈ C∞b (Rn), φi ∈ H, n ≥ 1 ⊂ L2(Ω)be the space of all “smooth cylindrical” random variables, where C∞b (Rd) denotesthe class of all bounded infinitely differentiable functions on Rn, whose partialderivatives are also bounded.

The Gross-Malliavin derivative of an element F = f(β(φ1, . . . , β(φn))) ∈Sβ , with respect to β, is defined by:

DβF :=n∑

i=1

∂f

∂xi(β(φ1), . . . , β(φn))φi ∈ L2(Ω;H).

We endow Sβ with the norm: ‖F‖2D1,2β

:= E|F |2 +E‖DβF‖2H, and let D1,2β be the

completion of Sβ with respect to this norm. The operator Dβ can be extended toD1,2

β . The adjoint δβ : Dom δβ ⊂ L2(Ω;H) → L2(Ω) of the operator Dβ , is calledthe Hitsuda-Skorohod integral with respect to β. The operator δβ is uniquelydefined by the following relation:

E(Fδβ(u)) = E〈DβF, u〉H, ∀F ∈ D1,2β .

We use the notation δβ(u) =∫ T

0usδβs. Note that E(δβ(u)) = 0, ∀u ∈ Dom δβ .

If V is a Hilbert space, let Sβ(V ) be the class of all “smooth cylindrical” V -valued random variables:

Sβ(V ) := u =m∑

j=1

Fjφj ;Fj ∈ Sβ , φj ∈ V,m ≥ 1 ⊂ L2(Ω;V ).

The Gross-Malliavin derivative of u =∑m

j=1 Fjφj ∈ SB(V ) is:

Dβu :=m∑

j=1

(DβFj)φj ∈ L2(Ω;H⊗ V ).

50 RALUCA BALAN

We endow SB(V ) with the norm:

‖u‖2D1,2β (V )

:= E‖u‖2V + E‖Dβu‖2H⊗V , (2.3)

and let D1,2β (V ) be the completion of Sβ(V ) with respect to this norm. The

operator Dβ can be extended to D1,2β (V ).

In particular, if V = H, then D1,2β (H) ⊂ Dom δβ , and

E|δβ(u)|2 = E‖u‖2H + E(〈Dβu, (Dβu)∗〉H⊗H))

≤ E‖u‖2H + E‖Dβu‖2H⊗H = ‖u‖2D1,2β (H)

, ∀u ∈ D1,2β (H) (2.4)

where (Dβu)∗ is the adjoint of Dβu in H ⊗ H. If u ∈ D1,2β (H) then Dβu ∈

L2(Ω;H⊗H). By abuse of notation, we write Dβu = (Dβt us)s,t∈[0,T ], even if Dβu

is not a function in s, t. As in [2], we introduce the following subspaces of D1,2β (H):

D1,2β (|H|) := u ∈ D1,2

β (H);u ∈ |H| a.s., Dβu ∈ H ⊗ |H| a.s., and

‖u‖D1,2β (|H|) <∞,

L1,2H,β := u ∈ D1,2

β (|H|); ‖u‖L1,2H,β

<∞,

where ‖u‖2D1,2β (|H|) := E‖u‖2|H| + E‖Dβu‖2H⊗|H| and

‖u‖2L1,2H,β

:= E

∫ T

0

u2sds+ E

∫ T

0

(∫ T

0

|Dβt us|1/Hdt

)2H

ds.

From (2.1) and (2.2), we have:

‖u‖D1,2β (|H|) ≤ bH‖u‖L1,2

H,β, ∀u ∈ L1,2

H,β . (2.5)

From (2.4), (2.1) and (2.2), it follows that:

E|δβ(u)|2 ≤ ‖u‖2D1,2β (|H|), ∀u ∈ D1,2

β (|H|), (2.6)

E|δβ(u)|2 ≤ b2H(E‖u‖2L1/H([0,T ]) + E‖Dβu‖2L1/H([0,T ]2)), ∀u ∈ L1,2H,β (2.7)

If u ∈ L1,2H,β then u1[0,t] ∈ D1,2

β (|H|) for all t ∈ [0, T ] and we denote δ(u1[0,t]) =∫ t

0usδβs. The following maximal inequality has been proved in [2]:

E supt≤T

∣∣∣∣∫ t

0

usδβs

∣∣∣∣2

≤ CH,T ‖u‖2L1,2H,β

, ∀u ∈ L1,2H,β (2.8)

where CH,T is a constant which depends on H and T .Note that L1,2

H,β may not be a Banach space with respect to the norm ‖ · ‖L1,2H,β

.

The following definition introduces a complete subspace of L1,2H,β .

Definition 2.1. Let Sβ(E) be the class of processes ut =∑m

i=1 Fi1(ti−1,ti](t), t ∈[0, T ], with Fi ∈ Sβ and 0 ≤ t0 < . . . < tm ≤ T . We denote by L1,2

H,β the completionof Sβ(E) with respect to the norm ‖ · ‖L1,2

H,β.


In summary, we have:

L1,2H,β ⊂ L1,2

H,β ⊂ D1,2β (|H|) ⊂ D1,2

β (H) ⊂ Dom δβ ⊂ L2(Ω;H). (2.9)

3. Malliavin Calculus for Hn2 -Valued Variables

We let C∞0 be the space of infinitely differentiable functions on Rd, with compactsupport, D be the space of real-valued Schwartz distributions on C∞0 , and L2 bethe space of all square-integrable functions on Rd. Let n ∈ R be arbitrary (notnecessarily an integer). The fractional Sobolev space of index n is:

Hn2 := u ∈ D; (1−∆)n/2u ∈ L2,

with the norm given by: ‖u‖Hn2

:= ‖(1−∆)n/2u‖L2 . (See e.g. p. 187, [18] for thedefinition of (1−∆)n/2). For any u ∈ Hn

2 and φ ∈ C∞0 , we set:

(u, φ) :=∫

Rd

[(1−∆)n/2u](x) · [(1−∆)−n/2φ](x)dx.

By the Cauchy-Schwartz inequality, we have:

|(u, φ)| ≤ N‖u‖Hn2, (3.1)

where N = Nn,φ = ‖(1−∆)−n/2φ‖L2 is a constant depending on n and φ.Let β = (βt)t∈[0,T ] be a fractional Brownian motion of Hurst index H > 1/2,

defined on a probability space (Ω,F , P ). In the present work, we introduce ananalogue of the space L1,2

H,β for Hn2 -valued functions.

Definition 3.1. If U is an arbitrary Hilbert space, we let

D1,2β (|HU |) := g ∈ D1,2

β (HU ); g ∈ |HU | a.s., Dβg ∈ H ⊗ |HU | a.s., and

‖g‖D1,2β (|HU |) <∞,

L1,2H,β(U) := g ∈ D1,2

β (|HU |); ‖g‖L1,2H,β(U) <∞,

where ‖g‖2D1,2β (|HU |) := E‖g‖2|HU | + E‖Dβg‖2H⊗|HU | and

‖g‖2L1,2H,β(U)

:= E

∫ T

0

‖gs‖2Uds+ E

∫ T

0

(∫ T

0

‖Dβt gs‖1/H

U dt

)2H

ds. (3.2)

Note that if U = R, then D1,2β (|HU |) = D1,2

β (|H|) and L1,2H,β(U) = L1,2

H,β .Let Sβ(EU ) be the space of all processes g(t, ·) =

∑mi=1 Fi1(ti−1,ti](t)vi, t ∈ [0, T ],

with Fi ∈ Sβ , 0 ≤ t0 < . . . < tm ≤ T, vi ∈ U . Note that Sβ(EU ) is dense inD1,2

β (|HU |) and

‖u‖D1,2β (|HU |) ≤ bH‖u‖L1,2

H,β(U), ∀u ∈ L1,2H,β(U). (3.3)

In the present article, we work with the space L1,2H,β(Hn

2 ) Let Sβ(EC∞0 ) be theclass of processes g(t, ·) =

∑mi=1 Fi1(ti−1,ti](t)φi(·), t ∈ [0, T ], with Fi ∈ Sβ , 0 ≤

t0 < . . . < tm ≤ T and φi ∈ C∞0 . Note that Sβ(EC∞0 ) is dense in D1,2β (|HHn

2|).

52 RALUCA BALAN

Remark 3.2. Note that g ∈ L1,2H,β(Hn

2 ) implies that g ∈ Hn2 and Dβg ∈ Hn

2,H , where

Hn2 := L2(Ω× [0, T ],F × B([0, T ]);Hn

2 )Hn

2,H := L2(Ω× [0, T ],F × B([0, T ]);L1/H([0, T ];Hn2 ))

are stochastic spaces of Sobolev type. (We should emphasize that our definitionfor the space Hn

2 is different than the one found in [18], since we are using theproduct σ-field F × B([0, T ]) instead of the predictable σ-field P.) Moreover,

‖g‖2L1,2H,β(Hn

2 )= E

∫ T

0

‖gs‖2Hn2ds+ E

∫ T

0

(∫ T

0

‖Dβt gs‖1/H

Hn2dt

)2H

ds

= ‖g‖2Hn2

+ ‖Dβg‖2Hn2,H. (3.4)

In what follows, we examine some of the properties of a random function g ∈D1,2

β (|HHn2|). To simplify the writing, we denote by ∗ the missing t variable of such

a function, to distinguish it from the missing x variable, denoted by ·.We have the following preliminary estimates.

Lemma 3.3. If g ∈ D1,2β (|HHn

2|), then for any φ ∈ C∞0 , we have:

E‖(g(∗, ·), φ)‖2|H| ≤ N2E‖g‖2|HHn2| (3.5)

E‖(Dβg(∗, ·), φ)‖2H⊗|H| ≤ N2E‖Dβg‖2H⊗|HHn2|, (3.6)

where N = Nn,φ = ‖(1−∆)−n/2φ‖L2 is a constant depending on n and φ.

Proof. The result follows by (3.1). ¤

The next result shows that for an arbitrary function g ∈ D1,2β (|HHn

2|), the Gross

derivative commutes with the action of a test function φ ∈ C∞0 .

Proposition 3.4. If g ∈ D1,2β (|HHn

2|), then for any φ ∈ C∞0 , we have (g(∗, ·), φ) ∈

D1,2β (|H|), and

Dβ(g(∗, ·), φ) = (Dβg(∗, ·), φ), (3.7)‖(g(∗, ·), φ)‖D1,2

β (|H|) ≤ N‖g‖D1,2β (|HHn

2|), (3.8)


Proof. Case 1. Suppose that g ∈ Sβ(EC∞0 ). Say g(t, ·) =∑m

i=1 Fi1(ti,ti+1](t)φi

with Fi ∈ Sβ , 0 ≤ t1 < . . . < tm+1 ≤ T and φi ∈ C∞0 . Denote Ψi(t, ·) =1(ti,ti+1](t)φi(·). Let φ ∈ C∞0 be arbitrary. Clearly (Ψi(∗, ·), φ) ∈ E , and hence(g(∗, ·), φ) =

∑mi=1 Fi(Ψi(∗, ·), φ) ∈ Sβ(E) ⊂ D1,2

β (|H|). Due to the linearity of Dβ ,we have: Dβ

t (g(s, ·), φ) =∑m

i=1(Dβt Fi)(Ψi(s, ·), φ) = (Dβ

t g(s, ·), φ). Finally, (3.8)follows from (3.7) and the preliminary estimates (3.5), (3.6):

‖(g(∗, ·), φ)‖2D1,2β (|H|) = E‖(g(∗, ·), φ)‖2|H| +E‖Dβ(g(∗, ·), φ)‖2H⊗|H|

= E‖(g(∗, ·), φ)‖2|H| +E‖(Dβg(∗, ·), φ)‖2H⊗|H|≤ N2E‖g‖2|HHn

2| +N2E‖Dβg‖2H⊗|HHn

2|

= N2‖g‖2D1,2β (|HHn

2|).


Case 2. Suppose that g ∈ D1,2β (|HHn

2|) is arbitrary. Then, there exists a

sequence (gj)j ⊂ Sβ(EC∞0 ) such that ‖gj − g‖D1,2β (|HHn

2|) → 0 as j → ∞, i.e.

E‖gj − g‖2|HHn2| → 0 and E‖Dβgj −Dβg‖2H⊗|HHn

2| → 0.

From Case 1, it follows that for any φ ∈ C∞0 , we have:

Dβ(gj(∗, ·), φ) = (Dβgj(∗, ·), φ). (3.9)

On one hand, due to the estimates (3.5) and (3.6), we have:

E‖(gj(∗, ·), φ)− (g(∗, ·), φ)‖2|H| ≤ N2E‖gj − g‖2|HHn2| → 0 (3.10)

E‖(Dβgj(∗, ·)−Dβg(∗, ·), φ)‖2H⊗|H| ≤ N2E‖Dβgj −Dβg‖2H⊗|HHn2|

→ 0. (3.11)

On the other hand, due to the estimate (3.8) obtained in Case 1, it followsthat (gj(∗, ·), φ)j is a Cauchy sequence in D1,2

β (|H|). Hence, there exists hφ ∈D1,2

β (|H|) such that ‖(gj(∗, ·), φ)− hφ‖D1,2β (|H|) → 0, i.e.

E‖(gj(∗, ·), φ)− hφ‖2|H| → 0, and (3.12)

E‖Dβ(gj(∗, ·), φ)−Dβhφ‖2H⊗|H| → 0 (3.13)

From (3.10) and (3.12), it follows that (g(∗, ·), φ) = hφ ∈ D1,2β (|H|). From (3.9),

(3.11) and (3.13), we conclude that (Dβg(∗, ·), φ) = Dβhφ = Dβ(g(∗, ·), φ), i.e.(3.7) holds. Based on (3.7), one deduces the estimate (3.8) as in Case 1. ¤

The next result is an immediate consequence of Proposition 3.4 and (3.1).

Corollary 3.5. If g ∈ L1,2H,β(Hn

2 ), then for any φ ∈ C∞0 , (g(∗, ·), φ) ∈ L1,2H,β, and

‖(g(∗, ·), φ)‖L1,2H,β

≤ N‖g‖L1,2H,β(Hn

2 ), (3.14)


The following definition introduces a complete subspace of ‖L1,2H,β(Hn

2 ).

Definition 3.6. We let L1,2H,β(Hn

2 ) be the completion of Sβ(EC∞0 ) with respect tothe norm ‖ · ‖L1,2

H,β(Hn2 ).

To summarize, here are the spaces introduced in this section:

L1,2H,β(Hn

2 ) ⊂ L1,2H,β(Hn

2 ) ⊂ D1,2β (|HHn

2|) ⊂ D1,2

β (HHn2) ⊂ L2(Ω;HHn

2).

4. The Infinite Dimensional Noise

Let us now consider a sequence βk = (βkt )t∈[0,T ], k ≥ 1 of i.i.d. fBm’s with

Hurst index H > 1/2, defined on the same probability space (Ω,F , P ).The following result generalizes the second-moment maximal inequality (2.8) to

an infinite sequence of i.i.d. fBm’s.

54 RALUCA BALAN

Theorem 4.1. Let uk ∈ L1,2H,βk be such that

∑∞k=1 ‖uk‖2L1,2

H,βk

<∞. Then

E supt≤T

∣∣∣∣∣∞∑

k=1

∫ t

0

uksδβ

ks

∣∣∣∣∣

2

≤ CH,T

∞∑

k=1

‖uk‖2L1,2H,βk

,

where CH,T is a constant depending on H and T .

Proof. As in the proof of Theorem 4, [2], we let α = 1/2− ε with ε ∈ (0,H − 1/2)and we use the fact that

∫ t

0uk

sδβks = cα

∫ t

0(t−r)−α

(∫ r

0uk

s(r − s)α−1δβks

)dr. Using

Cauchy-Schwartz inequality and the fact that 2α < 1, we obtain∣∣∣∣∣∞∑

k=1

∫ t

0

uksδβ

ks

∣∣∣∣∣

2

= c2α

∣∣∣∣∣∫ t

0

(t− r)−α

( ∞∑

k=1

∫ r

0

uks(r − s)α−1δβk

s

)dr

∣∣∣∣∣

2

≤ c2α

(∫ t

0

(t− r)−2αdr

) ∫ t

0

∣∣∣∣∣∞∑

k=1

∫ r

0


s

∣∣∣∣∣

2

dr

= c′α

∫ t

0

∣∣∣∣∣∞∑

k=1

∫ r

0


s

∣∣∣∣∣

2

dr, ∀t ≤ T.

Therefore,

supt≤T

∣∣∣∣∣∞∑

k=1

∫ t

0

uksδβ

ks

∣∣∣∣∣

2

≤ c′α

∫ T

0

∣∣∣∣∣∞∑

k=1

∫ r

0


s

∣∣∣∣∣

2

dr. (4.1)

Let vks = uk

s(r − s)α−1, s ∈ [0, T ] and note that vk ∈ L1,2H,βk . Since (βk)k are

independent fBm’s and each vk ∈ L1,2H,βk , it follows that the random variables

Xk =∫ r

0vk

s δβks , k ≥ 1 are independent. Moreover, E(Xk) = 0 for all k. Hence

E

∣∣∣∣∣n∑

k=1

∫ r

0


s

∣∣∣∣∣

2

=n∑

k=1

E

∣∣∣∣∫ r

0


s

∣∣∣∣2

. (4.2)

Using the Fatou’s lemma, we infer that

E

∣∣∣∣∣∞∑

k=1

∫ r

0


s

∣∣∣∣∣

2

≤∞∑

k=1

E

∣∣∣∣∫ r

0


s

∣∣∣∣2

. (4.3)

From (4.1) and (4.3), we get:

E supt≤T

∣∣∣∣∣∞∑

k=1

∫ t

0

uksδβ

ks

∣∣∣∣∣

2

≤ c′α

∞∑

k=1

∫ T

0

E

∣∣∣∣∫ r

0


s

∣∣∣∣2

dr.

Using (2.7), we get

E supt≤T

∣∣∣∣∣∞∑

k=1

∫ t

0

uksδβ

ks

∣∣∣∣∣

2

≤ c′αb2H

∞∑

k=1

∫ T

0

E

(∫ r

0

|uks |1/H(r − s)(α−1)/Hds

)2H

dr+


∞∑

k=1

∫ T

0

E

(∫ r

0

∫ T

0

|Dβk

θ uks |1/H(r − s)(α−1)/Hdθds

)2H

dr

:= c′αb

2H(I1 + I2).

Using Holder’s inequality with p = 2H and q = 2H/(2H − 1), we get:(∫ r

0

|uks |1/H(r − s)(α−1)/Hds

)2H

≤ cα,Hr2(α−1)+2H−1

∫ r

0

|uks |2ds,

(∫ r

0

∫ T

0

|Dβk

θ uks |1/H(r − s)(α−1)/Hdθds

)2H

≤

cα,Hr2(α−1)+2H−1

∫ r

0

(∫ T

0

|Dβk

θ uks |1/Hdθ

)2H

ds,

and hence

I1 ≤ cα,H

∫ T

0

r2(α−1)+2H−1∞∑

k=1

E

∫ r

0

|uks |2dsdr ≤ cα,H,TE

∞∑

k=1

∫ T

0

|uks |2ds

I2 ≤ cα,H

∫ T

0

r2(α−1)+2H−1∞∑

k=1

E

∫ r

0

(∫ T

0

|Dβk

θ uks |1/Hdθ

)2H

dsdr

≤ cα,H,TE

∞∑

k=1

∫ T

0

(∫ T

0

|Dβk

θ uks |1/Hdθ

)2H

ds.

¤

In what follows, we let l2 be the set of all real-valued sequences a = (ak)k with∑k |ak|2 <∞.The following space is the l2-variant of the space L1,2

H,β(Hn2 ).

Definition 4.2. We denote by L1,2H (Hn

2 , l2) the set of all g = (gk)k such thatgk ∈ D1,2

βk (|HHn2|) for all k and ‖g‖L1,2

H (Hn2 ,l2)

<∞, where

‖g‖2L1,2H (Hn

2 ,l2):=

∞∑

k=1

‖gk‖2L1,2H,βk (Hn

2 ). (4.4)

We also consider the following l2-variants of the stochastic spaces Hn2 and Hn

2,H ,defined in Remark 3.2:

Hn2 (l2) := g = (gk)k; gk ∈ Hn

2 ∀k, ‖g‖Hn2 (l2) <∞

Hn2,H(l2) := g = (gk)k; gk ∈ Hn

2,H ∀k, ‖g‖Hn2,H(l2) <∞,

where

‖g‖2Hn2 (l2)

:=∞∑

k=1

‖gk‖2Hn2

and ‖g‖2Hn2,H(l2)

:=∞∑

k=1

‖gk‖2Hn2,H.

56 RALUCA BALAN

Let Dg := (Dβk

gk)k be the “Gross derivative” of g = (gk)k ∈ L1,2H (Hn

2 , l2). Then,

‖g‖2L1,2H (Hn

2 ,l2)= ‖g‖2Hn

2 (l2)+ ‖Dg‖2Hn

2,H(l2)(4.5)

∞∑

k=1

E

∫ T

0

‖gk(s, ·)‖2Hn2ds+ E

∫ T

0

(∫ T

0

‖Dβk

t gk(s, ·)‖1/HHn

2dt

)2H

ds

.(4.6)

The following definition introduces the space in which we are allowed to pickthe random coefficient g = (gk)k, multiplying the noise (βk)k.

Definition 4.3. We let L1,2H (Hn

2 , l2) be the set of all g ∈ L1,2H (Hn

2 , l2) for whichthere exists a sequence (gj)j ⊂ L1,2

H (Hn2 , l2) such that ‖gj − g‖L1,2

H (Hn2 ,l2)

→ 0 asj →∞, (gk

j )j ⊂ Sβk(EC∞0 ) for k ≤ Kj , and gkj = 0 for k > Kj .

Theorem 4.4. Let g ∈ L1,2H (Hn

2 , l2) be arbitrary. Then g ∈ L1,2H (Hn

2 , l2) if andonly if gk ∈ L1,2

H,βk(Hn2 ) for all k.

Proof. The argument is standard and is omitted. ¤

5. The Solution Space

The following definition introduces the solution space for the stochastic heatequation, whose noise term is given by a sequence of i.i.d. fBm’s.

Definition 5.1. Let βk = (βkt )t∈[0,T ], k ≥ 1 be a sequence of i.i.d. fBm’s with

Hurst index H > 1/2, defined on the same probability space (Ω,F , P ).Let u = u(t, ·)t∈[0,T ] be a D-valued random process defined on the probability

space (Ω,F , P ). We write u ∈ Hn2,H if:

(i) u(0, ·) ∈ L2(Ω,F ;Hn−12 );

(ii) u ∈ Hn2 , uxx ∈ Hn−2

2 ; and(iii) there exist f ∈ Hn−2

2 and g ∈ L1,2H (Hn−1

2 , l2) such that for any φ ∈ C∞0 ,the equality

(u(t, ·), φ) = (u(0, ·), φ) +∫ t

0

(f(s, ·), φ)ds+∞∑

k=1

∫ t

0

(gk(s, ·), φ)δβks (5.1)

holds for any t ∈ [0, T ] a.s. We define

‖u‖Hn2,H

= (E‖u(0, ·)‖2Hn−1

2)1/2 + ‖uxx‖Hn−2

2+ ‖f‖Hn−2

2+ ‖g‖L1,2

H (Hn−12 ,l2)

. (5.2)

The next lemma shows that the series of stochastic integrals in (5.1) convergesuniformly in t ∈ [0, T ], in probability.

Lemma 5.2. Let g ∈ L1,2H (Hn

2 , l2) and φ ∈ C∞0 be arbitrary. For each t ∈ [0, T ],let X(K)

t =∑K

k=1

∫ t

0(gk(s, ·), φ)δβk

s ,K ≥ 1, and Xt =∑∞

k=1

∫ t

0(gk(s, ·), φ)δβk

s .Then Xt is finite a.s., and the sequence (X(K))K converges in probability to X, inthe sup-norm metric, i.e. limK→∞ P (supt≤T |X(K)

t −Xt| ≥ ε) = 0, ∀ε > 0.


Proof. Let uks = (gk(s, ·), φ). Note that uk ∈ L1,2

H,βk for all k. By Theorem 4.1 and(3.14), E(X2

t ) ≤ CH,T

∑∞k=1 ‖uk‖2L1,2

H,βk

≤ CH,TN2∑∞

k=1 ‖gk‖2L1,2H,βk (Hn

2 )<∞.

Hence, Xt is finite a.s. Using Chebyshev’s inequality, Theorem 4.1, and (3.14), weget:

P (supt≤T

|X(K)t −Xt| ≥ ε) ≤ 1

ε2E sup

t≤T

∣∣∣∣∣∞∑

k=K+1

∫ t

0

uksδβ

ks

∣∣∣∣∣

2

≤

1ε2CH,T

∞∑

k=K+1

‖uks‖2L1,2

H,βk

≤ 1ε2CH,TN

2∞∑

k=K+1

‖gk‖2L1,2H,βk (Hn

2 ).

The last terms converge to 0 as K →∞, since ‖g‖L1,2H (Hn

2 ,l2)<∞. ¤

Remark 5.3. Note that by Theorem 5, [2], each process X(K) has an a.s. con-tinuous modification. By invoking the previous lemma, and using a “classical”argument in probability theory (see e.g. the proof of Theorem 6.1.10, [19]), weconclude that the process X has an a.s. continuous modification. By (5.1), it fol-lows that if u ∈ Hn

2,H , then the process (u(t, ·), φ)t∈[0,T ] has an a.s. continuousmodification, for any φ ∈ C∞0 . We work with this modification.

For technical reasons, one prefers not to handle directly the elements in the spaceHn

2 , and work instead with the images of these elements in the nicer space L2, viathe operator (1 − ∆)n/2. More liberty in choosing the right index n dependingon the problem at hand, comes from the fact that the operator (1−∆)m/2 mapsisometrically Hn

2 onto Hn−m2 , for any n and m. This property continues to hold

for the stochastic spaces Hn2 and Hn

2 (l2) (see Remark 3.4, [18]). The next tworesults empower us with the same freedom of choice of the right index n, whenworking with the newly introduced spaces L1,2

H,β(Hn2 ) and L1,2

H (Hn2 , l2), and the

solution space Hn2,H .

Proposition 5.4. The operator (1−∆)m/2 maps isometrically L1,2H (Hn

2 , l2) ontoL1,2

H (Hn−m2 , l2).

Proof. Step 1. Let β = (βt)t∈[0,T ] be a fixed fBm. We first prove that (1−∆)m/2

maps isometrically L1,2H,β(Hn

2 ) onto L1,2H,β(Hn−m

2 ).Let g ∈ L1,2

H,β(Hn2 ) be arbitrary. We prove that (1 − ∆)m/2g ∈ L1,2

H,β(Hn−m2 ).

Since the Malliavin derivative commutes with the action of a test function φ ∈ C∞0(see (3.7)), we have:

(Dβ [(1−∆)m/2g(∗, ·)], φ) = Dβ((1−∆)m/2g(∗, ·), φ) =

Dβ(g(∗, ·), (1−∆)m/2φ) = (Dβg(∗, ·), (1−∆)m/2φ) =

((1−∆)m/2[Dβg(∗, ·)], φ),

for any φ ∈ C∞0 , i.e.

Dβt [(1−∆)m/2g(s, ·)] = (1−∆)m/2[Dβ

t g(s, ·)], ∀s, t ∈ [0, T ]. (5.3)

58 RALUCA BALAN

Using an approximation argument and the fact that ‖u‖Hn2

= ‖(1−∆)m/2u‖Hn−m2

for any u ∈ Hn2 , we conclude that (1−∆)m/2g ∈ D1,2

β (|HHn−m2

|). Using (3.4) and(5.3), we obtain:

‖(1−∆)m/2g‖2L1,2H,β(Hn−m

2 )= ‖(1−∆)m/2g‖2Hn−m

2+ ‖Dβ [(1−∆)m/2g]‖2Hn−m

2,H

= ‖g‖2Hn2

+ ‖Dβg‖2Hn2,H

= ‖g‖2L1,2H,β(Hn

2 ). (5.4)

Let g ∈ L1,2H,β(Hn

2 ) be arbitrary and h := (1 − ∆)m/2g. An approximationargument shows that h ∈ L1,2

H,β(Hn−m2 ). More precisely, we know that there exists

a sequence (gj)j ⊂ Sβ(EC∞0 ) such that ‖gj − g‖L1,2H,β(Hn

2 ) → 0 as j →∞. Note that

hj := (1−∆)m/2gj ∈ Sβ(EC∞0 ) for any j. By (5.4), we have ‖hj −h‖L1,2H,β(Hn−m

2 ) =

‖gj − g‖L1,2H,β(Hn

2 ) → 0 as j →∞. Hence, h ∈ L1,2H,β(Hn−m

2 ).

Finally, if h ∈ L1,2H,β(Hn−m

2 ) is arbitrary, we let g = (1 − ∆)−m/2h; then g ∈L1,2

H,β(Hn2 ) and (1−∆)m/2g = h. This proves that (1−∆)m/2 is onto.

Step 2. Let βk = (βkt )t∈[0,T ], k ≥ 1 be a sequence of i.i.d fBm’s. We now prove

that (1−∆)m/2 maps isometrically L1,2H (Hn

2 , l2) onto L1,2H (Hn−m

2 , l2).Let g = (gk)k ∈ L1,2

H (Hn2 , l2) be arbitrary. By Step 1,

(1−∆)m/2gk ∈ D1,2βk (|HHn−m

2|)

and ‖(1−∆)m/2gk‖L1,2H,βk (Hn−m

2 ) = ‖gk‖L1,2H,βk (Hn

2 ), for all k. Hence

‖(1−∆)m/2g‖L1,2H (Hn−m

2 ,l2)= ‖g‖L1,2

H (Hn2 ,l2)

<∞,

and (1 − ∆)m/2g ∈ L1,2H (Hn−m

2 , l2). The fact that (1 − ∆)m/2 is onto follows bythe same principles as in Step 1.

Step 3. Finally, let g = (gk)k ∈ L1,2H (Hn

2 , l2) be arbitrary. Then, there existsa sequence (gj)j ⊂ L1,2

H (Hn2 , l2) such that ‖gj − g‖L1,2

H (Hn2 ,l2)

→ 0 and (gkj )j ⊂

Sβk(EC∞0 ) for each k. Using Step 2, ‖(1−∆)m/2gj − (1−∆)m/2g‖L1,2H (Hn−m

2 ,l2)=

‖gj − g‖L1,2H (Hn

2 ,l2)→ 0. Since (1−∆)m/2gk

j j ⊂ Sβk(EC∞0 ) for each k, it follows

that (1−∆)m/2g ∈ L1,2H (Hn−m

2 , l2). ¤

Corollary 5.5. The operator (1−∆)m/2 maps isometrically Hn2,H onto Hn−m

2,H .

Proof. The argument is the same as in Remark 3.4, [18], and is omitted. ¤

The following definition introduces the deterministic and stochastic componentsof a solution process u.

Definition 5.6. If u ∈ Hn2,H and (5.1) holds for some f ∈ Hn−2

2 and g = (gk)k ∈L1,2

H (Hn−12 , l2), then we write du = fdt +

∑∞k=1 g

kδβkt , t ∈ [0, T ]. We say that

Du := f is the deterministic part of u, and Su = (Sku)k := g is the stochasticpart of u.


Remark 5.7. The operators D : Hn2,H → Hn−2

2 and S : Hn2,H → L1,2

H (Hn−12 , l2) are

continuous, by the definition of the norm in Hn2,H .

The next theorem is the analogue of Theorem 3.7, [18], whose proof we followvery closely. The essential difference is that we use the maximal inequality givenby Theorem 4.1, instead of the Burkholder-Davis-Gundy inequality.

Theorem 5.8. (a) For any u ∈ Hn2,H , we have

E supt≤T

‖u(t, ·)‖2Hn−2

2≤ N‖u‖2Hn

2,H, and (5.5)

‖u‖Hn2≤ N‖u‖Hn

2,H, (5.6)

where N is a constant which depends on d, T and H.(b) The space Hn

2,H is a Banach space with the norm (5.2).

Proof. We refer the reader to the proof of Theorem 3.7, [18] for the details.(a) By Proposition 5.4, it suffices to consider only the case n = 2. We want

to prove that E supt≤T ‖u(t, ·)‖2L2≤ N‖u‖2H2

2,Hfor any u ∈ H2

2,H . This can beachieved via the Fatou’s lemma, once we show that

E supt≤T

‖u(ε)(t, ·)‖2L2≤ N‖u‖2H2

2,H, ∀ε > 0 (5.7)

supt≤T

‖u(1/m)(t, ·)− u(t, ·)‖L2 → 0 as m→∞, a.s. (5.8)

(Here u(ε) = u ∗ ζε is the “mollification” of u using a test function ζ ∈ C∞0 , ζ ≥ 0with

∫Rd ζ(x)dx = 1, and ζε(x) = ε−dζ(x/d).)

To prove (5.7), we note that u(ε) satisfies (5.1) with the pair (f (ε), g(ε)) in placeof (f, g) (see (3.6) of [18]). The estimates for u(ε)(0, ·) and f (ε) are the same as in[18]. The estimate for g(ε) is obtained using a different technique. More precisely,using Theorem 4.1, for all x ∈ Rd we have

E supt≤T

∣∣∣∣∣∞∑

k=1

∫ t

0

g(ε)k(s, x)δβks

∣∣∣∣∣

2

≤ CH,T

∞∑

k=1

‖g(ε)k(∗, x)‖2L1,2H,βk

.

We integrate with respect to x. Using Minkowksi’s inequality and the fact that‖h(ε)‖L2 ≤ ‖h‖L2 for any h ∈ L2, we get

E supt≤T

∥∥∥∥∥∞∑

k=1

∫ t

0

g(ε)k(s, x)δβks

∥∥∥∥∥

2

L2

≤ CH,T

∞∑

k=1

E

∫ T

0

∫

Rd

|g(ε)k(s, x)|2dxds

+∞∑

k=1

E

∫ T

0

∫

Rd

(∫ T

0

|Dβk

θ g(ε)k(s, x)|1/Hdθ

)2H

dxds

≤ CH,T

∞∑

k=1

E

∫ T

0

∥∥gk(s, ·)∥∥2

L2ds+ E

∫ T

0

(∫ T

0

‖Dβk

θ gk(s, ·)‖1/HL2

dθ

)2H

ds

= CH,T ‖g‖2L1,2H (L2,l2)

≤ CH,T ‖g‖2L1,2H (H1

2 ,l2)≤ CH,T ‖u‖2H2

2,H.

60 RALUCA BALAN

(We used the fact that Dβk

θ g(ε)k(s, x) = (Dβk

θ gk(s, x))(ε), which is a consequenceof (3.7), since g(ε)k(s, x) = (gk(s, ·), ζε(x − ·)).) The arguments for proving (5.8)and (b) are similar to those of [18] and are omitted. ¤

6. The Existence and Uniqueness of a Solution

In this section, we consider the stochastic heat equation:

du(t, x) = (∆u(t, x) + f(t, x))dt+∞∑

k=1

gk(t, x)δβkt , t ∈ [0, T ]. (6.1)

This equation is interpreted in the sense of Definition 5.6. More precisely, we saythat u ∈ Hn

2,H is a solution of (6.1) if Du = ∆u+ f and Su = g.The next theorem is the main result of the present paper, which can be viewed

as an analogue of Theorem 4.2, [18].

Theorem 6.1. Let n ∈ R be arbitrary. Let

f ∈ Hn2 and g ∈ L1,2

H (Hn+12 , l2).

Then, equation (6.1) with zero initial condition has a unique solution u ∈ Hn+22,H .

For this solution, we have

‖uxx‖Hn2≤ N(‖f‖Hn

2+ ‖g‖L1,2

H (Hn+12 ,l2)

), and (6.2)

‖u‖Hn+22,H

≤ N(‖f‖Hn2

+ ‖g‖L1,2H (Hn+1

2 ,l2)), (6.3)

where N is a constant depending on d, T and H.

Some preliminaries are needed before we can give the proof of this result. Recallthat if f(t, x) and u0(x) are deterministic functions, then a solution of: ut =∆u + f , u(0, ·) = u0, is given by u(t, x) = Ttu0(x) +

∫ t

0Tt−s[f(s, ·)](x)ds, where

Tth(x) = (4πt)−d/2∫Rd h(y)e−|x−y|2/(4t)dy.

The following result is due to Doyoon Kim (personal communication).

Lemma 6.2. We have:∫ t

rTt−s(∆φ)(x)ds = Tt−rφ(x)− φ(x), for all φ ∈ C∞0 .

Proof. Let z be the solution of zt = ∆z, z(0, ·) = φ(·), and z be the solutionof zt = ∆z, z(0, ·) = ∆φ(·). Then z(t, x) = Ttφ(x) and z(t, x) = Tt(∆φ)(x) =∆[Ttφ(x)] = ∆z(t, x). Hence∫ t

r

Tt−s(∆φ)(x)ds =∫ t

r

z(t− s, x)ds =∫ t

r

∆z(t− s, x)ds =∫ t

r

zt(t− s, x)ds

= z(t− r, x)− z(0, x) = Tt−rφ(x)− φ(x).

¤

The first idea of the proof is to treat separately the particular case when thegk’s are smooth elementary processes (in which case the solution can be writtenin closed form), and then apply an approximation argument.

The second idea is to evaluate (in norm) the difference between the solution uof the original equation (6.1) and the solution u1 of the “deterministic” equation(i.e. equation (6.1) with gk = 0 for all k), having in mind that bounds for u1 areavailable from the PDE theory. This is achieved by the following proposition.


Proposition 6.3. Let f ∈ H−12 and u1(t, x) =

∫ t

0Tt−s[f(s, ·)](x)ds. Let gk ∈

Sβk(EC∞0 ) for k ≤ K, and gk = 0 for k > K. Let u(t, x) = v(t, x) +∫ t

0Tt−s[(∆v+

f)(s, ·)](x)ds, where v(t, x) =∑∞

k=1

∫ t

0gk(s, x)δβk

s . Then

‖u− u1‖H02

≤ N‖g‖L1,2H (L2,l2)

(6.4)

‖ux − u1x‖H02

≤ N‖g‖L1,2H (L2,l2)

(6.5)

‖uxx − u1xx‖H−12

≤ N‖g‖L1,2H (L2,l2)

, (6.6)

where N is a constant depending on d, T and H.

Proof. Let gk(t, ·) =∑mk

i=1 Fki 1(tk

i−1,tki ](t)gk

i (·), with F ki ∈ Sβk , 0 ≤ tk0 < . . . <

tkmk≤ T (non-random) and gk

i ∈ C∞0 .We begin with the proof of (6.4). By definition, u(t, x) − u1(t, x) = v(t, x) +∫ t

0Tt−s[∆v(s, ·)](x)ds. Note that

v(s, x) =∞∑

k=1

∫ s

0

gk(r, x)δβkr =

∞∑

k=1

mk∑

i=1

gki (x)

∫ s

0

F ki 1(tk

i−1,tki ](r)δβ

kr , (6.7)

and hence Tt−s[∆v(s, ·)](x) =∑∞

k=1

∑mk

i=1 Tt−s(∆gki )(x)

∫ s

0F k

i 1(tki−1,tk

i ](r)δβkr . By

the stochastic Fubini’s theorem and Lemma 6.2, it follows that:

u(t, x)− u1(t, x) = v(t, x) +∞∑

k=1

mk∑

i=1

∫ t

0

Tt−s(∆gki )(x)

∫ s

0

F ki 1(tk

i−1,tki ](r)δβ

kr ds

= v(t, x) +∞∑

k=1

mk∑

i=1

∫ t

0

F ki 1(tk

i−1,tki ](r)

∫ t

r

Tt−s(∆gki )(x)dsδβk

r

= v(t, x) +∞∑

k=1

mk∑

i=1

∫ t

0

F ki 1(tk

i−1,tki ](r)(Tt−rg

ki (x)− gk

i (x))δβkr .

Using (6.7) and the fact that

Tt−r[gk(r, ·)](x) =mk∑

i=1

F ki 1(tk

i−1,tki ](r)Tt−rg

ki (x), (6.8)

we obtain that: u(t, x)−u1(t, x) =∑∞

k=1

∫ t

0Tt−r[gk(r, ·)](x)δβk

r . From here, usingTheorem 4.1 and Fubini’s theorem, we get:

‖u− u1‖2H02

=∫ T

0

∫

Rd

E

∣∣∣∣∣∞∑

k=1

∫ t

0

Tt−s[gk(s, ·)](x)δβks

∣∣∣∣∣

2

dxdt

≤ CH,T

∞∑

k=1

E

∫ T

0

∫ t

0

∫

Rd

∣∣Tt−s[gk(s, ·)](x)∣∣2 dxdsdt+

∞∑

k=1

E

∫ T

0

∫ t

0

∫

Rd

(∫ T

0

|Dβk

θ Tt−s[gk(s, ·)](x)|1/Hdθ

)2H

dxdsdt

:= CH,T (I1 + I2).


and hence

∫ T

0

∫

Rd

1s≤t|Dβk

θ Tt−s[gk(s, ·)](x)|2dxdt

=mk∑

i=1

|Dβk

θ F ki |21(tk

i−1,tki ](s)]

∫ T

s

∫

Rd

|F(Tt−sgki )(ξ)|2dξdt

=mk∑

i=1

|Dβk

θ F ki |21(tk

i−1,tki ](s)

∫

Rd

|Fgki (ξ)|2

(∫ T

s

e−(t−s)|ξ|2dt

)dξ

≤ NT

mk∑

i=1

|Dβk

θ F ki |21(tk

i−1,tki ](s)

∫

Rd

|gki (x)|2dx

= NT

∫

Rd

∣∣∣∣∣mk∑

i=1

(Dβk

θ F ki )1(tk

i−1,tki ](s)g

ki (x)

∣∣∣∣∣

2

dx

= NT

∫

Rd

|Dβk

θ gk(s, x)|2dx = NT ‖Dβk

θ gk(s, ·)‖2L2,

where we used (6.9) for the second equality above, and (6.10) for the inequality.From here, we obtain that

I2 ≤ NT

∞∑

k=1

E

∫ T

0

(∫ T

0

‖Dβk

θ gk(s, ·)‖1/HL2

dθ

)2H

ds = NT ‖Dg‖2H02,H(l2)

. (6.12)

Relation (6.4) follows by taking the sum of (6.11) and (6.12), and using (4.5).We now turn to the proof of (6.5). Note that ux(t, x) − u1x(t, x) = vx(t, x) +∫ t

0Tt−s[∆vx(s, ·)](x)ds =

∑∞k=1

∫ t

0Tt−r[gk

x(r, ·)](x)δβkr , and

Tt−s[gkx(s, ·)](x) =

mk∑

i=1

F ki 1(tk

i−1,tki ](s)Tt−sg

kix(x), (6.13)

where we use the notation Tt−sgkix(x) = (Tt−sg

kixl

(x))1≤l≤d and gkixl

= ∂gki /∂xl.

By Theorem 4.1, we get:

‖ux − u1x‖2H02

=∫ T

0

∫

Rd

E

∣∣∣∣∣∞∑

k=1

∫ t

0

Tt−s[gkx(s, ·)](x)δβk

s

∣∣∣∣∣

2

dxdt

≤ CH,T

∞∑

k=1

E

∫ T

0

∫ t

0

∫

Rd

∣∣Tt−s[gkx(s, ·)](x)∣∣2 dxdsdt+

∞∑

k=1

E

∫ T

0

∫ t

0

∫

Rd

(∫ T

0

|Dβk

θ Tt−s[gkx(s, ·)](x)|1/Hdθ

)2H

dxdsdt

:= CH,T (J1 + J2).

64 RALUCA BALAN

We treat J1 first. Using (6.13), we get:

∫

Rd

|Tt−s[gkx(s, ·)](x)|2dx =

mk∑

i=1

|F ki |21(tk

i−1,tki ](s)

∫

Rd

|Tt−sgkix(x)|2dx

=mk∑

i=1

|F ki |21(tk

i−1,tki ](s)

∫

Rd

|ξ|2e−(t−s)|ξ|2 |Fgki (ξ)|2dξ,

where the second equality is due to the fact that for l = 1, . . . , d,

F(Tt−sgkixl

)(ξ) = iξlF(Tt−sgki )(ξ) = iξle

−(t−s)|ξ|2Fgki (ξ), (6.14)

which can be proved using integration by parts and (6.9).Using Fubini’s theorem and the fact that

∫ T

s

e−(t−s)|ξ|2dt ≤ 1− e−T |ξ|2

|ξ|2 ≤ 1|ξ|2 , ∀ξ ∈ Rd, (6.15)

we obtain:

J1 =∞∑

k=1

E

∫ T

0

∫ t

0

mk∑

i=1

|F ki |21(tk

i−1,tki ](s)

∫

Rd

|ξ|2e−(t−s)|ξ|2 |Fgki (ξ)|2dξdsdt

=∞∑

k=1

E

∫ T

0

mk∑

i=1

|F ki |21(tk

i−1,tki ](s)

∫

Rd

(∫ T

s

e−(t−s)|ξ|2dt

)|ξ|2|Fgk

i (ξ)|2dξds

≤∞∑

k=1

E

∫ T

0

mk∑

i=1

|F ki |21(tk

i−1,tki ](s)

∫

Rd

|Fgki (ξ)|2dξds

=∞∑

k=1

E

∫ T

0

∫

Rd

|gk(s, x)|2dxds = ‖g‖2H02(l2)

. (6.16)

We treat J2 next. Using Minkowski’s inequality, we get:

J2 =∞∑

k=1

E

∫ T

0

∫ T

0

∫

Rd

(∫ T

0

1s≤t|Dβk

θ Tt−s[gkx(s, ·)](x)|1/Hdθ

)2H

dxdtds

≤∞∑

k=1

E

∫ T

0

∫ T

0

(∫ T

s

∫

Rd

|Dβk

θ Tt−s[gkx(s, ·)](x)|2dxdt

)1/(2H)

dθ

2H

ds.

By (6.13), we have

|Dβk

θ Tt−s[gkx(s, ·)](x)|2 =

mk∑

i=1

|Dβk

θ F ki |21(tk

i−1,tki ](s)|Tt−sg

kix(x)|2,


and therefore,∫ T

0

∫

Rd

1s≤t|Dβk

θ Tt−s[gkx(s, ·)](x)|2dxdt

=mk∑

i=1

|Dβk

θ F ki |21(tk

i−1,tki ](s)

∫ T

s

∫

Rd

|F(Tt−sgkix)(ξ)|2dξdt

=mk∑

i=1

|Dβk

θ F ki |21(tk

i−1,tki ](s)

∫

Rd

|ξ|2Fgki (ξ)

(∫ T

s

e−(t−s)|ξ|2dt

)dξ

≤mk∑

i=1

|Dβk

θ F ki |21(tk

i−1,tki ](s)

∫

Rd

|gki (x)|2dx

=∫

Rd

∣∣∣∣∣mk∑

i=1

(Dβk

θ F ki )1(tk

i−1,tki ](s)g

ki (x)

∣∣∣∣∣

2

dx = ‖Dβk

θ gk(s, ·)‖2L2,

where we used (6.14) for the second equality above, and (6.15) for the inequality.From here, we obtain that

J2 ≤∞∑

k=1

E

∫ T

0

(∫ T

0

‖Dβk

θ gk(s, ·)‖1/HL2

dθ

)2H

ds = ‖Dg‖2H02,H(l2)

. (6.17)

Relation (6.5) follows by taking the sum of (6.16) and (6.17), and using (4.5).Finally, (6.6) follows from (6.5) since ‖uxx − u1xx‖H−1

2≤ N‖ux − u1x‖H0

2. ¤

We are now ready to give the proof of the main result.

Proof. (of Theorem 6.1) By Proposition 5.4, it suffices to take n = −1.Case 1. Suppose that gk ∈ Sβk(EC∞0 ) for k ≤ K and gk = 0 for k > K.

Let u(t, x) = v(t, x) + z(t, x), where v is as in Proposition 6.3, and z satisfiesz(t, x) =

∫ t

0(∆z + ∆v + f)(s, x)ds =

∫ t

0(∆u + f)(s, x)ds. Clearly, u is a solution

of (6.1).We now check that u ∈ H1

2,H , i.e. u satisfies (i)-(iii) of Definition 5.1. Sinceu(0, ·) = 0, (i) holds. Also, (iii) holds with Du = ∆u+ f and Su = g. It remainsto check (ii). Let u1(t, x) =

∫ t

0Tt−s[f(s, ·)](x)ds. From the PDE theory,

‖u1‖L2([0,T ],L2) ≤ N‖f‖L2([0,T ],H−12 ) (6.18)

‖u1xx‖L2([0,T ],H−12 ) ≤ N‖f‖L2([0,T ],H−1

2 ), (6.19)

where N is a constant depending on d and T . Using (6.18) and (6.4), we get:

‖u‖H02

≤ ‖u1‖H02+ ‖u− u1‖H0

2

≤ N(‖f‖H−12

+ ‖g‖L1,2H (L2,l2)

). (6.20)

Using (6.19) and (6.6), we get:

‖uxx‖H−12

≤ ‖u1xx‖H−12

+ ‖uxx − u1xx‖H−12

≤ N(‖f‖H−12

+ ‖g‖L1,2H (L2,l2)

). (6.21)

66 RALUCA BALAN

Using the fact that ‖φ‖H12≤ ‖φ‖L2 + ‖φxx‖H−1

2, (6.20) and (6.21), we get:

‖u‖H12

≤ ‖u‖H02+ ‖uxx‖H−1

2

≤ 2N(‖f‖H−12

+ ‖g‖L1,2H (L2,l2)

).

From here, we conclude that u ∈ H12 and uxx ∈ H−1

2 , i.e. u verifies condition (ii)of Definition 5.1. Since Du = ∆u+ f , we also infer that

‖u‖H12,H

= ‖uxx‖H−12

+ ‖∆u+ f‖H−12

+ ‖g‖L1,2H (L2,l2)

≤ N(‖f‖H−12

+ ‖g‖L1,2H (L2,l2)

)).

The uniqueness of the solution of (6.1) in H12,H follows from the uniqueness of

the solution of the classical heat equation.

Case 2. Let g = (gk)k ∈ L1,2H (L2, l2) be arbitrary. By Theorem 4.4, there exists

a sequence (gj)j ⊂ L1,2H (L2, l2) such that ‖gj − g‖L1,2

H (L2,l2)→ 0, gk

j ∈ Sβk(EC∞0 )for k ≤ Kj and gk

j = 0 for k > Kj .Using the result proved in Case 1, we know that there exists a unique solution

uj ∈ H12,H of the equation

duj(t, x) = (∆uj + f)(t, x)dt+∞∑

k=1

gkj (t, x)δβk

t , t ∈ [0, T ], (6.22)

with zero initial condition. This solution satisfies:

‖ujxx‖H−12≤ N(‖f‖H−1

2+ ‖gj‖L1,2

H (L2,l2)), (6.23)

‖uj‖H12,H

≤ N(‖f‖H−12

+ ‖gj‖L1,2H (L2,l2)

). (6.24)

From here, it follows that (uj)j is a Cauchy sequence inH12,H . By Theorem 5.8.(b),

there exists u ∈ H12,H such that ‖uj − u‖H1

2,H→ 0.

We now prove that u is a solution of (6.1). Since ‖uj − u‖H12,H

→ 0 and the

operators D : H12,H → H−1

2 and S : H12,H → L1,2

H (L2, l2) are continuous, it followsthat

‖Duj −Du‖H−12→ 0 and ‖Suj − Su‖L1,2

H (L2,l2)→ 0. (6.25)

On the other hand, from (6.22) it follows that Duj = ∆uj +f and Suj = gj . Since‖ujxx − uxx‖H−1

2≤ ‖uj − u‖H1

2,H→ 0, we get that Duj → ∆u+ f in H−1

2 . Hence

‖Duj − (∆u+ f)‖H−12→ 0 and ‖Suj − g‖L1,2

H (L2,l2)→ 0. (6.26)

From (6.25) and (6.26), we infer that Du = ∆u + f and Su = g, i.e. u satisfies(6.1). Finally, (6.2) and (6.3) are obtained by passing to the limit in (6.23) and(6.24). ¤

Acknowledgment. The author would like to thank Professor Hui-Hsiung Kuoand an anonymous referee for their comments, which led to an improvement of themanuscript.


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68 RALUCA BALAN

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Raluca Balan: Department of Mathematics and Statistics, University of Ottawa,Ottawa, ON, K1N 6N5, Canada


URL: http://aix1.uottawa.ca/∼rbalan

ON THE DISTRIBUTIONS OF THE SUP AND INF OF THE

CLASSICAL RISK PROCESS WITH EXPONENTIAL CLAIM*

JORGE A. LEON AND JOSE VILLA

Abstract. The purpose of this article is to use the double Laplace transformof the occupation measure of the classical risk process X with exponentialclaim to deduce the distributions of the random variables supXs : s ≤ tand infXs : s ≤ t, for every t > 0. As a consequence, we also get thedistributions of the time to ruin in finite time and the first passage of a givenlevel.

1. Introduction

In this paper, we deal with the classical risk process with exponential claimdefined on a complete probability space (Ω,F , P ). More precisely, let

Xt = x0 + ct −Nt∑

k=1

Rk, t ≥ 0. (1.1)

Here x0 ≥ 0 is the initial capital, c > 0 is the premium income per unit oftime, N = Nt, t ≥ 0 is an homogeneous Poisson process with rate λ and Rk,k = 1, 2, . . . is a sequence of i.i.d. random variables independent of N . Henceforthwe suppose that R1 has exponential distribution with mean 1/r.

Our goal is to calculate explicit expressions for the distributions of the randomvariables supXs : s ≤ t and infXs : s ≤ t, t > 0. Toward this end, we applythe complex inversion theorem of the Laplace transform (or Lerch’s theorem) tothe double Laplace transforms of some occupation measures of X (see Section 2below). As a consequence, we are also able to give the distribution of the firstpassage of certain level x ∈ R of the process X . It means, the distributions of

Sx = inft > 0 : Xt = x and Tx = inft > 0 : Xt < x,because the right-continuity of the process X yields

Sx < t =

sups≤t

Xs > x

, x > x0

and

Tx ≤ t =

infs≤t

Xs < x

, x < x0.

2000 Mathematics Subject Classification. Primary 60K30; Secondary 60K99.Key words and phrases. Classical risk process, Laplace transform, ruin probability, time to

ruin.* Partially supported by a sabbatical year grant of CONACyT and PIM 08-2 of UAA.

69



70 JORGE A. LEON AND JOSE VILLA

The study of the distribution of the time to ruin T0 in finite time (i.e., for thecase x = 0) is extensive in the literature on risk theory due to its applicationsin business activities. For instance, numerical procedures have been utilized byseveral authors in the analysis of T0 (see, for example, Dickson and Waters [6] orSeal [12]). This numerical approximations have been improved in several works byderiving expressions for the mentioned distribution (see Asmussen [1, 2], Dickson[4], Dickson et al. [5, 7], Drekic and Willmot [9], and Ignatov and Kaishev [11],among others). In particular, the method used in [5], [7] and [9] is based on thecomplex inversion theorem, as we does here.

For x ∈ R, the process Sx have been analyzed by doss Reis [8] and Gerber [10]via a martingale method.

The paper is organized as follows. In Section 2 we relate the Laplace transformsof the functions

P

sups≤•

Xs ≤ x

and P

infs≤•

Xs ≤ x

(1.2)

to the double Laplace transform of some occupation measures of X . Then, inSection 3 we use the complex inversion theorem to calculate the two probabilitiesin (1.2).

2. Occupation Measures

Now we are interested in the Laplace transforms of the occupation measures

Yx(t) =

∫ t

0

1(x,+∞)(Xs)ds and Y x(t) =

∫ t

0

1(−∞,x)(Xs)ds,

with x ∈ R and t > 0. So, we assume that the reader is familiar with the elementaryproperties of the Laplace transform as they are presented, for example, in Spiegel[13].

Throughout, the Laplace transform of a measurable function h : [0,∞) → R isdenoted by L(h). That is,

L(h)(s) =

∫ ∞

0

e−sth(t)dt,

for s ∈ R such that this integral is convergent.The relation between the occupation measures Yx and Y x, and the probabilities

in (1.2) is given by the following result.

Proposition 2.1. Let X be the classical risk process defined in (1.1) and x ∈ R.Then for each t > 0,

Yx(t) = 0 =

sups≤t

Xs ≤ x

and Y x(t) = 0 =

infs≤t

Xs ≥ x

.

Proof. We first observe that Yx(t) = 0 ⊃

sups≤t Xs ≤ x

is trivial. Now wesee the reverse inclusion. Let ω ∈ Ω be such that

∫ t

0

1(x,+∞)(Xs(ω))ds = 0. (2.1)

CLASSICAL RISK PROCESS 71

If there is some s0 ∈ (0, t) such that Xs0(ω) > x, then, by the right-continuityof X , there exists a non-empty open interval Is0 ⊂ (0, t) such that s0 ∈ Is0 andXs(ω) > x, for all s ∈ Is0 . Consequently,

∫ t

0

1(x,+∞)(Xs(ω))ds ≥∫

Is0

1(x,+∞)(Xs(ω))ds = |Is0 | > 0,

where |Is0 | is the length of Is0 . But this is a contradiction to (2.1). ThereforeXs(ω) ≤ x for all s ≤ t, which implies that ω also belongs to

sups≤t Xs ≤ x

.We proceed similarly for the remainder of the proof.

In order to express the double Laplace transform of Y x and Yx, we need tointroduce the following notation. Let s be a positive real number. The positiveand negative roots of the quadratic equation

cv2 + (rc − λ − s)v − sr = 0

are denoted by v+s and v−s , respectively.

Proposition 2.2. Let s and α be two positive real numbers. Then

∫ ∞

0

e−stE[

e−αYx(t)]

dt =

1s+α

1 +α( 1

c−v+

ss )

v−

s+α−v+s −α

c

e(x0−x)v−

s+α

, x0 ≥ x,

1s +

αscs+α ( 1

c−v+

ss )v−

s+α−α(v−

s+α−v+s −α

c )sc(v+

s + αc )(v−

s+α−v+s −α

c )e(x−x0)v+s

, x0 < x,

and

∫ ∞

0

e−stE[

e−αY x(t)]

dt =

1s+α

1 +α( 1

c −v−

ss )

v+s+α−v−

s −αc

e(x0−x)v+s+α

, x0 ≤ x,

1s +

αscs+α ( 1

c−v−

ss )v+

s+α−α(v+s+α−v−

s −αc )

sc(v−

s + αc )(v+

s+α−v−

s −αc )e(x−x0)v

−

s, x0 ≥ x.

Proof. For a < b and t > 0 define the double Laplace transform of T[a,b](t) =∫ t

01[a,b](Xr)dr as

f(x0) =

∫ ∞

0

e−stEx0

[

e−αT[a,b](t)]

dt.

This is a Feynman-Kac representation of the solution of equation

Af(x0) + 1 =

(s + α)f(x0), a < x0 < b,sf(x0), x0 < a or x0 > b,

where A is the infinitesimal generator associated to the semigroup of process X .Solving this equation and letting a ↓ −∞ and b ↑ ∞, respectively, we are done.For details see the paper of Chiu and Yin [3] (Corollary 4.1).

The following result is a consequence of Proposition 2.2 and it will be used inSection 3.

Proposition 2.3. Let X be the classical risk process given by (1.1). Then, forevery s > 0, we have

L(P (Yx(·) = 0))(s) =

0, x0 ≥ x,1s − e−(x−x0)v+

s

s , x0 < x,(2.2)


and

L(P (Y x(·) = 0))(s) =

0, x0 ≤ x,1s − λe−(x−x0)v−

s

cs(v+s +r)

, x0 > x.(2.3)

Proof. We first deal with equality (2.2). By the dominated convergence theoremwe can write

limα→∞

∫ ∞

0

e−stE[

e−αYx(t)]

dt

= limα→∞

∫ ∞

0

e−st

(

∫

Yx(t)=0+

∫

Yx(t)>0

)

e−αYx(t)dPdt

=

∫ ∞

0

e−stP (Yx(t) = 0) dt +

∫ ∞

0

e−st

∫

Yx(t)>0

(

limα→∞

e−αYx(t))

dPdt

=

∫ ∞

0

e−stP (Yx(t) = 0) dt.

Therefore, from Proposition 2.2 we get∫ ∞

0

e−stP (Yx(t) = 0) dt (2.4)

= limα→∞

1s+α

1 +α( 1

c −v+

ss )

v−

s+α−v+s −α

c

e(x0−x)v−

s+α

, x0 ≥ x,

1s +

αscs+α ( 1

c−v+

ss )v−

s+α−α(v−

s+α−v+s −α

c )

sc(v+s + α

c )(v−

s+α−v+s −α

c )e(x−x0)v+s

, x0 < x.

Note that the definition of v−s+α implies that limα→∞(

v−s+α/α)

= 0, which leadsus to

limα→∞

(

α(

v−s+α − v+s − α

c

)−1)

= −c.

Hence, the fact that v−s+α < 0, for all α > 0, together with (2.4), yields thatequality (2.2) is true for x0 ≥ x.

On the other hand, for x0 < x,

limα→∞

1

s+

αscs+α

(

1c − v+

s

s

)

v−s+α − α(


c

)

sc(

v+s + α

c

) (


c

)

e(x−x0)v+s

= limα→∞

1

s+

sc(

1c − v+

s

s

)

v−

s+α

s+α −(


c

)

sc(

v+s

α + 1c

)

(


c

)

e−(x−x0)v+s

=1

s− e−(x−x0)v

+s

s.

Thus (2.2) holds.Finally, in order to see that (2.3) is satisfied, we only need to observe that

limα→∞

(

v+s+α/α

)

=1

cand lim

α→∞

(

v+s+α − v−s − α

c

)

= v+s + r,

and proceed as in the beginning of this proof.


3. The Distributions of the Sup and Inf of X Within Finite Time

The purpose of this section is to apply Lerch’s theorem to the Laplace trans-forms obtained in Proposition 2.3 to calculate the distributions of the sup and infof X (see Proposition 2.1).

In order to state the main result of this paper, we need to introduce the followingnotation.

Note first that

(s + λ − rc)2 + 4crs = (s + λ + rc)2 − 4λrc

= (s + λ + rc − 2√

λrc)(s + λ + rc + 2√

λrc)

= (s − r1)(s − r2),

with r1 = 2√

λrc − λ − rc and r2 = −2√

λrc − λ − rc. Hence r2 < r1 < 0,

v+s =

s + λ − rc +√

(s − r1)(s − r2)

2c, (3.1)

and

v−s =s + λ − rc −

√

(s − r1)(s − r2)

2c. (3.2)

For sake of simplicity, let us utilize the conventions a = x−x0

2c and b = λ + rc.

Theorem 3.1. Let X be the classical risk process (1.1) and t > 0. Then we have

P

(

sups≤t

Xs > x

)

(3.3)

= e−a(λ−rc)

(

e−a√

r1r2 +1

π

∫ r1

r2

e(t−a)u sin(a |u − r1|1/2 |u − r2|1/2)

udu

)

,

for every x ∈ (x0, x0 + ct), and

P

(

infs≤t

Xs < x

)

= 2λe−a(λ−rc)

(

ea√

r1r2

b +√

r1r2(3.4)

+1

πIm

(

∫ r1

r2

e(t−a)u−ai|u−r1|1/2|u−r2|1/2

u(

u + b − i|u − r1|1/2|u − r2|1/2)du

))

,

for every x < x0.

Remark 3.2. We make two remarks.

i) Note that

P

(

sups≤t

Xs > x

)

= 0 for x ≥ x0 + ct, and P

(

sups≤t

Xs > x

)

= 1 for x < x0.

ii) Similarly we have

P

(

infs≤t

Xs < x

)

= 1 for x > x0.

In the following two subsections, we separate the proofs of (3.3) and (3.4) forthe convenience of the reader because both of them are long and tedious.


3.1. Proof of equality (3.3). Let hM (s) = e−as−a√

(s−r1)(s−r2)

s , s ∈ C\[r2, r1],

with√

(s − r1)(s − r2) = |(s − r1)(s − r2)|1/2 exp(iχ(s)), where

χ(s) =arg(s − r1) + arg(s − r2)

2.

Then, by (2.2), (3.1), the inverse theorem of the Laplace transform (see for example[13]) and the fact that hM is an analytic function on C\[r2, r1], we have, for σ largeenough,

P (Yx(t) = 0) =1

2πi

∫ σ+i∞

σ−i∞ets

(

1

s− e−(x−x0)v

+s

s

)

ds

= 1 − 1

2πi

∫ σ+i∞

σ−i∞ets ea(rc−λ−s−

√(s−r1)(s−r2))

sds

= 1 − ea(rc−λ)

2πi

∫ σ+i∞

σ−i∞etshM (s)ds.

Notice that we can use the inverse theorem because it is not difficult to see thatt 7→ P (Xt = x) = 0 is continuous, which follows from the fact that P (Xt = x) = 0.

Let C(ρ, ε) = C1 ∪ · · · ∪ C13 be the following contour of integration:

B

A

r1r2σ

ρ

C1C2

C3

C4

C5 C6

C7

C8C9

C10

C11

C12 C13

ǫ

ρ − η(ǫ)

6

-

Now observe that 0 is a pole of order one and r1, r2 are branch points of hM .Therefore, by the residue theorem (see [13]),

∫ σ+i∞

σ−i∞etshM (s)ds = 2πie−a

√r1r2 − lim

ρ→∞limε→0

∫

C(ρ,ε)

etshM (s)ds. (3.5)


Note that we only need to analyze the integral in the right-hand side of (3.5)on each arc Cj , j ∈ 1, . . . , 13, in order to finish the proof. To do so, now wedivide the proof in several steps.

Step 1. We begin our study on the arcs C1(ρ) and C13(ρ).For C1(ρ) we take the parametrization s = ρeiθ, θ1

1 ≤ θ ≤ π/2, where ρ and θ11

are indicated in the following figure:

r1r2 σ

ρ

ρ cos θ

C1

θ11

θ

6

-

In this case, it is easy to see that

0 ≤ arg(ρeiθ − r1) <π

2and 0 ≤ arg(ρeiθ − r2) <

π

2,

which give 0 ≤ χ(ρeiθ) < π/2. Since cos θ ≥ 0, when θ ∈ [0, π/2], we have

Re

(

√

(ρeiθ − r1)(ρeiθ − r2)

)

= |ρeiθ − r1|1/2|ρeiθ − r2|1/2 cos(χ(ρeiθ)) > 0.

Using this and the fact that a > 0, we can conclude

e−aRe

“√(ρeiθ−r1)(ρeiθ−r2)

”

≤ 1. (3.6)

Now note that t > a, due to x0 + ct > x, and that ρ cos θ < σ. Thus,∣

∣

∣

∣

∣

∫

C1(ρ)

etshM (s)ds

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∫ π2

θ11

e(t−a)ρeiθ−a√

(ρeiθ−r1)(ρeiθ−r2)

ρeiθρieiθdθ

∣

∣

∣

∣

∣

≤∫ π

2

θ11

e(t−a)ρ cos θ−aRe


”

dθ

≤∫ π

2

θ11

e(t−a)σdθ

= e(t−a)σ sin−1

(

σ

ρ

)

≤ e(t−a)σ

(

σ

ρ

)

π

2.

Hence limρ→∞∫

C1(ρ)etshM (s)ds = 0.

We can proceed in the same way to see that limρ→∞∫

C13(ρ)etshM (s)ds is also

equal to zero.


Note that an important point in this analysis is the fact that t − a > 0 and−a < 0. This will be also important in the remaining of this proof.

Step 2. Now we consider the integral over C2(ρ) and C12(ρ).Over C2(ρ), we consider the parametrization s = ρeiθ, π/2 ≤ θ ≤ θ2

2 < π, as itis ilustrated in the next figure:

r1r2

C2

θ22

θ

ρ

6

-

As in the previous case, the inequality (3.6) is still true. So, taking into accountthat sin θ ≥ 2θ/π, for θ ∈ [0, π/2], we conclude

∣

∣

∣

∣

∣

∫

C2(ρ)

etshM (s)ds

∣

∣

∣

∣

∣

≤∫ θ2

2

π2



”

dθ

≤∫ θ2

2

π2

e(t−a)ρ cos θdθ

=

∫ θ22−π

2

0

e(t−a)ρ cos(θ+ π2 )dθ

=

∫ θ22−π

2

0

e−(t−a)ρ sin θdθ

≤∫ θ2

2−π2

0

e−(t−a)ρ(2θ/π)dθ ≤ π

2ρ(t − a).

From which we get

limρ→∞

∫

C2(ρ)

etshM (s)ds = 0.

Similarly,

limρ→∞

∫

C12(ρ)

etshM (s)ds = 0.


Step 3. Here we will show that

limρ→∞

∫

C3(ρ)

etshM (s)ds = 0

and

limρ→∞

∫

C11(ρ)

etshM (s)ds = 0.

On C3(ρ), we still use the parametrization s = ρeiθ, with π2 < θ2

2 ≤ θ ≤ θ33 < π:

r1r2

C3

θ33

θ22

θ

ρ

6

-

Since x < x0 + ct, then t > 2a. Therefore we can take η > 0 such that

t >(

1 + (1 + η)1/4)

a. (3.7)

Moreover take ρ > 0 such that

ηρ2 > |r2 − r1|2 . (3.8)

Notice that

π

2≤ arg(ρeiθ − r1) < θ and 0 ≤ arg(ρeiθ − r2) ≤

π

2.

Hence

π

4≤ χ(ρeiθ) <

θ + π2

2< θ.

Since cos θ is decreasing on [π/4, π], we have

Re

(

√


)

≥∣

∣ρeiθ − r1

∣

∣

1/2 ∣∣ρeiθ − r2

∣

∣

1/2cos θ. (3.9)

On the other hand, the fact that r1 ≥ ρ cos θ implies∣

∣ρeiθ − r1

∣

∣ ≤ ρ. (3.10)


And using (3.8), we get

∣

∣ρeiθ − r2

∣

∣ ≤√

|r1 − r2|2 + ρ2 sin2(θ)

≤√

|r1 − r2|2 + ρ2

≤ (η + 1)1/2

ρ. (3.11)

Therefore (3.9), (3.10) and (3.11) yields

Re

(

√


)

≥ ρ1/2 (η + 1)1/4

ρ1/2 cos θ (3.12)

= (η + 1)1/4 ρ cos θ.

From this and (3.7) we obtain∣

∣

∣

∣

∣

∫

C3(ρ)

etshM (s)ds

∣

∣

∣

∣

∣

≤∫ θ3

3

θ22



”

dθ

≤∫ θ3

3

θ22

e(t−a)ρ cos θ−a(η+1)1/4ρ cos θdθ

=

∫ θ33

θ22

e(t−a(1+(η+1)1/4))ρ cos θdθ

=

∫ θ33−π

2

θ22−π

2

e(t−a(1+(η+1)1/4))ρ cos(θ+π2 )dθ

=

∫ θ33−π

2

θ22−π

2

e−(t−a(1+(η+1)1/4))ρ sin θdθ

≤∫ θ3

3−π2

θ22−π

2

e−(t−a(1+(η+1)1/4))ρ 2θπ dθ

≤ π

2ρ(t − a(1 + (η + 1)1/4

)).

This implies that

limρ→∞

∫

C3(ρ)

etshM (s)ds = 0.

Proceeding as the beginning of this step, we can conclude that we also have

limρ→∞

∫

C11(ρ)

etshM (s)ds = 0.

Step 4. Now we deal with the arcs C4(ρ, ε) and C10(ρ, ε).Here we consider the same parametrization of previous steps. That is, s =

ρeiθ, π/2 ≤ θ33 ≤ θ ≤ θ4

4 ≤ π:


r1r2

C4

θ33

ρ − η(ǫ)

θ44

θ

ρ

6

-

Notice that

π

2≤ arg(ρeiθ − r1) ≤ θ and

π

2≤ arg(ρeiθ − r2) ≤ θ.

Thus, π/2 ≤ χ(ρeiθ) ≤ θ ≤ π. Moreover since r1, r2 ≥ ρ cos θ then

∣

∣ρeiθ − r1

∣

∣ ≤ ρ and∣

∣ρeiθ − r2

∣

∣ ≤ ρ,

and using the monotony of cos on [π/2, π], we have

Re

(

√


)

= |ρeiθ − r1|1/2|ρeiθ − r2|1/2 cos(χ(ρeiθ))

≥ |ρeiθ − r1|1/2|ρeiθ − r2|1/2 cos θ

≥ ρ cos θ.

The above estimation is analogous to (3.12). Now the conclusion follows as inStep 3.

Step 5. Here we consider C5(ρ, ε) and C9(ρ, ε).Over C5(ρ, ε) we take the parametrization s = u + εi, −ρ + η(ε) ≤ u ≤ r2 :

r1r2

C5ε

0−ρ + η(ε)-

Notice that

π

2≤ arg(u + εi − r1) ≤ π and

π

2≤ arg(u + εi − r2) ≤ π,


then π/2 ≤ χ(u + εi) ≤ π. Since cosine is negative and decreasing on [π/2, π] wehave, for ε < |r1 − r2|,

− aRe(

√

(u + εi − r1)(u + εi − r2))

= a|u + εi − r1|1/2|u + εi − r2|1/2(− cos(χ(u + εi)))

≤ 21/2a |u − r1| .

Hence

∣

∣etshM (s)∣

∣ ≤ e(t−a)u−aRe

“√(u+εi−r1)(u+εi−r2)

”

|u + εi| (3.13)

≤ e(t−a)r2+21/2a|ρ−r1|

|r2|.

By (3.13) we can apply the dominated convergence theorem and since

arg(u + εi − r1) → π and arg(u + εi − r2) → π, as ε → 0,

we have

limε→0

∫

C5(ρ,ε)

etshM (s)ds = limε→0

∫ r2

−ρ+η(ε)

e(t−a)(u+εi)−a√

(u+εi−r1)(u+εi−r2)

u + εidu

=

∫ r2

−ρ

e(t−a)u−a|u−r1|1/2|u−r2|1/2eiπ

udu

=

∫ r2

−ρ

e(t−a)u+a|u−r1|1/2|u−r2|1/2

udu.

On the other hand, on C9(ρ, ε) we use the parametrization s = −u− εi, −r2 ≤u ≤ ρ − η(ε) :

r1r2

C9

0

−ε

−ρ + η(ε)-

Working as in previous case, and noting that

arg(−u − εi − r1) → π and arg(−u − εi − r2) → π, as ε → 0,


we have

limε→0

∫

C9(ρ,ε)

etshM (s)ds

= limε→0

∫ ρ−η(ε)

−r2

e(t−a)(−u−εi)−a√

(−u−εi−r1)(−u−εi−r2)

−u − εi(−du)

= −∫ r2

−ρ

e(t−a)u−a|u−r1|1/2|u−r2|1/2eiπ

udu

= −∫ r2

−ρ

e(t−a)u+a|u−r1|1/2|u−r2|1/2

udu.

Therefore,

limρ→∞

(

limε→0

∫

C5(ρ,ε)

etshM (s)ds + limε→0

∫

C9(ρ,ε)

etshM (s)ds

)

= 0.

Step 6. Now we deal with C6(ε). To do this, we take the parametrizations = u + εi, r2 ≤ u ≤ r1 :

r1r2

C6

0-

Notice that for ε < 1,

|etshM (s)| =

∣

∣

∣

∣

∣



u + εi

∣

∣

∣

∣

∣

=e(t−a)u−a|u+εi−r1|1/2|u+εi−r2|1/2 cos χ(u+εi)

|u + εi|

≤ e(t−a)r1e−a|u+εi−r1|1/2|u+εi−r2|1/2 cos χ(u+εi)

|r1|

≤ e(t−a)r1ea|u+εi−r1|1/2|u+εi−r2|1/2

|r1|

≤ e−(t−a)r1+a(1+|r1−r2|2)1/2

|r1|. (3.14)

Since

arg(u + εi − r1) → π and arg(u + εi − r2) → 0, as ε → 0,


then√

(u + εi − r1)(u + εi − r2) = |u + εi − r1|1/2 |u + εi − r2|1/2exp (iχ(u + εi))

→ |u − r1|1/2 |u − r2|1/2i, as ε → 0.

Due to (3.14) we are able to apply the dominated convergence theorem:

limε→0

∫

C6(ε)

etshM (s)ds = limε→0

∫ r1

r2



u + εidu

=

∫ r1

r2

e(t−a)u−a|u−r1|1/2|u−r2|1/2i

udu.

Step 7. Now we suppose that C8(ε) is defined by s = −u−εi, −r1 ≤ u ≤ −r2 :

r1r2

C8

0-

we can imitate Step 6 to we get

limε→0

∫

C8(ε)

etshM (s)ds = −∫ r1

r2

e(t−a)u+a|u−r1|1/2|u−r2|1/2i

udu.

Step 8. Finally we deal with C7(ε). Here, we put s = r1 + εe−iθ, −π/2 ≤ θ ≤π/2 :

r1r2 0

θ

εC7

-

Thus

χ(εe−iθ + r1) ∈[

3

2π, 2π

]

∪[

0,1

2π

]

.

Consequently, using the fact that cos θ ≥ 0, for θ ∈ [32π, 2π] ∪ [0, 12π], we obtain

Re

(

√

εe−iθ(εe−iθ + r1 − r2)

)

= ε1/2|εe−iθ + r1 − r2|1/2 cosχ(εe−iθ + r1) ≥ 0.

(3.15)


Also, for 0 < ε < −r1/2, we have

∣

∣r1 + εe−iθ∣

∣ =√

r21 + 2r1ε cos(−θ) + ε2

≥ |r1 + ε| = −r1 − ε > −r1

2. (3.16)

The estimations (3.15), (3.16) and t > a yield∣

∣

∣

∣

∣

∫

C7(ε)

etshM (s)ds

∣

∣

∣

∣

∣

≤ εe(t−a)r1

∫ π2

−π2

∣

∣

∣

∣

∣

e(t−a)εe−iθ−a√

εe−iθ(εe−iθ+r1−r2)

r1 + εe−iθ

∣

∣

∣

∣

∣

dθ

≤ 2εe(t−a)r1

|r1|

∫ π2

−π2

∣

∣

∣e(t−a)εe−iθ−a

√εe−iθ(εe−iθ+r1−r2)

∣

∣

∣dθ

=2εe(t−a)r1

|r1|

∫ π2

−π2

e(t−a)ε cos(−θ)−aRe

“√εe−iθ(εe−iθ+r1−r2)

”

dθ

≤ 2εe(t−a)(r1+ε)

|r1|π.

Therefore limε→0

∫

C7(ε)etshM (s)ds = 0.

Step 9. To finish the proof, we only need to take into account Steps 1-8,together with (3.5), and Propositions 2.1 and 2.3.

3.2. Proof of (3.4). Let us define

K(s) = s + b +√

(s − r1)(s − r2), s ∈ C\[r2, r1].

Since

2b + r1 + r2 = 0 6= −4λrc = r1r2 − b2,

then

(s + b)2 6= (s − r1)(s − r2), ∀s ∈ C.

This implies that 1/K is analytic over C\[r2, r1]. Moreover, it is not difficult tosee that, for ρ large enough, we have

|K(s)| ≥

2√

λrc, s ∈ C2 ∪ C4 ∪ C5 ∪ C7 ∪ C9 ∪ C10 ∪ C12,b, s ∈ C1 ∪ C13,(ρ + r1)

1/2(ρ + r2)1/2 sin π

4 , s ∈ C3 ∪ C11,

ε + |Re(s) − r1|1/2 |Re(s) − r2|1/2 sin π4 , s ∈ C6 ∪ C8.

(3.17)

As in the proof of (3.3) we have by (3.2)

P (Y x(t) = 0) = 1 − 1

2πi

∫ σ+i∞

σ−i∞ets λe−(x−x0)v

−

s

cs(

v+s + r

) ds

= 1 − λe−a(λ−rc)

πi

∫ σ+i∞

σ−i∞etshm(s)ds,


where

hm(s) =e−as+a

√(s−r1)(s−r2)

sK(s).

Finally, the result follows from Proposition 2.1, (3.17) and the proof of (3.3).Observe that a < 0 and t− 2a > 0, together with (3.17), allow us to copy, line byline, the proof of (3.3) to show that (3.4) is also true.

Acknowledgment. The first author would like to thank Universidad Autonomade Aguascalientes and the second author appreciates the hospitality of Cinvestav-IPN and Universidad Juarez Autonoma de Tabasco during the realization of thiswork.

References

1. Asmussen, S.: Approximations for the probability of ruin within finite time, Scand. Act. J.

1 (1984) 31-57.2. Asmussen, S.: Ruin Probabilities, World Scientific Publishing Co., Singapure, 2000.3. Chiu, S. N., Yin, C.: On occupation times for a risk process with reserve-dependent premium,

Stochastic Models 18(2) (2002) 245-255.4. Dickson, D. C. M.: Some finite time ruin problems, Research Paper Series of Faculty of

Economics & Commerce, The University of Melbourne 153 (2007).5. Dickson, D. C. M., Hughes, B.D., Lianzeng, Z.: The density of the time to ruin for a Sparre

Andersen process with Erlang arrivals and exponential claims, Scan. Actuarial J. 5 (2005)358-376.

6. Dickson, D. C. M., Waters, H. R.: The probability and severity of ruin in finite and in infinitetime, Astin Bulletin 22(2) (1992) 177-190.

7. Dickson, D. C. M., Willmot, G. E.: The density of the time to ruin in the classical Poissonrisk model, Astin Bulletin 35(1) (2005) 45-60.

8. dos Reis, A. E.: How long is the surplus below zero?, Insurance: Mathematics and Economics

12 (1993) 23-38.9. Drekic, S., Willmot, G. E.: On the density and moments of the time of ruin with exponential

claims, Astin Bulletin 33(1) (2003) 11-21.10. Gerber, H. U.: When does the surplus reach a given target?, Insurrace: Mathematics and

Economics 9 (1990) 115-119.11. Ignatov, Z. G., Kaishev, V. K.: A finite-time ruin probability formula for continuous claim

severities, J. Appl. Prob. 41 (2004) 570-578.12. Seal, H. L.: Numerical calculations of the probability of ruin in the Poisson/Exponential

case, Mitt. Verein Schweiz. Versich. Math. 72 (1972) 77-100.13. Spiegel, M. R.: Laplace Transforms, McGraw-Hill, New York, 1988.

Jorge A. Leon: Cinvestav-IPN, Departamento de Control Automatico, Apartado

Postal 14-740, Mexico D.F., Mexico


Jose Villa: Universidad Autonoma de Aguascalientes, Depart. de Matematicas y

Fısica, Av. Universidad 940, C.P. 20100 Aguascalientes, Ags., Mexico


AN INTERACTING FOCK SPACE CHARACTERIZATIONOF PROBABILITY MEASURES

LUIGI ACCARDI, HUI-HSIUNG KUO, AND AUREL I. STAN

Abstract. In this paper we characterize the probability measures, on Rd,with square summable support, in terms of their associated preservation op-erators and the commutators of the annihilation and creation operators.

1. Introduction

A program of expressing properties of a probability measure on Rd, having finitemoments of any order, in terms of their annihilation, creation, and preservationoperators, was initiated in [2]. There, it was proved that a probability measureis polynomially symmetric if and only if all of its preservation operators vanish.The notion of “polynomial symmetry” is a weak form of the notion of “symmetry”from classic Measure Theory, in the sense that a probability measure µ, on Rd, iscalled symmetric if, for any Borel subset B of Rd, µ(B) = µ(−B), where −B :=−x | x ∈ B, while µ is called polynomially symmetric if for any monomialxi1

1 xi22 · · ·xid

d , such that i1 + i2 + · · ·+ id is odd, we have∫Rd xi1

1 xi22 · · ·xid

d µ(dx) = 0,where x = (x1, x2, . . . , xd) ∈ Rd.

It was also proved in [2], that a probability measure µ on Rd, having finitemoments of any order, is polynomially factorisable, if and only if, for all 1 ≤ i <j ≤ d, any operator from the set a−(i), a0(i), a+(i) commutes with any operatorfrom the set a−(j), a0(j), a+(j), where, for any k ∈ 1, 2, . . . , d, a−(k),a0(k), and a+(k), denote the annihilation, preservation, and creation operatorsof index k, respectively. Again the notion of “polynomial factorisability” is aweak form of the notion of “product measure” from Measure Theory, since itdoes not necessarily mean that µ is a product measure of d probability measuresµ1, µ2, . . . , µd on R, but only the fact that, for any monomial xi1

1 xi22 · · ·xid

d ,∫Rd xi1

1 xi22 · · ·xid

d µ(dx) =∫Rd xi1

1 µ(dx)∫Rd xi2

2 µ(dx) · · · ∫Rd xid

d µ(dx).In [3], it was proved that two probability measures µ and ν, on Rd, having finite

moments of any order, have the same moments, if and only if they have the samepreservation operators and the same commutators between the annihilation andcreation operators. The domain of these operators is understood to be the spaceof all polynomial functions of d real variables x1, x2, . . . , xd, with complex coeffi-cients. Thus the whole information about the moments of a probability measureis contained in two families of operators, namely the preservation operators and

2000 Mathematics Subject Classification. Primary 60A10; Secondary 05E35, 47B25.Key words and phrases. Probability measure, annihilation operator, creation operator, preser-

vation operator, commutator, square summable support, Hilbert-Schmidt operator.

85



86 LUIGI ACCARDI, HUI-HSIUNG KUO, AND AUREL I. STAN

the commutators between the annihilation and creation operators. Hence, ratherthan considering the annihilation and creation operators separately, we can studyproperties of probability measures, having finite moments of any order, by lookingat the joint action of these operators, expressed in terms of their commutators.

In this paper we continue the program started in [2], in the spirit of [3], bycharacterizing the probability measures, on Rd, with square summable support, interms of their preservation operators and the commutators between the annihila-tion and creation operators. We regard the result, from this paper, as an exampleof the interesting applications of quantum probabilistic, more precisely interactingFock space, techniques, to the classical probability theory. We have included aminimal background about the notions of annihilation, preservation, and creationoperators in section 2. The definition of the probability measures with squaresummable support and the main result of this paper are presented in section 3.

2. Background

Let µ be a probability measure defined on the Borel sigma field B of Rd, whered is a fixed positive integer. Throughout this paper, we assume that µ has finitemoments of any order, which means that for any i ∈ 1, 2, . . . , d and any p > 0,∫Rd |xi|pµ(dx) < ∞, where xi denotes the ith coordinate of the d−dimensional

vector x = (x1, x2, . . . , xd) ∈ Rd. For any non–negative integer n, we denote byFn, the space of all polynomial functions p(x1, x2, . . . , xd), of d real variables x1,x2, . . . , xd, with complex coefficients, and of total degree less than or equal ton. In Fn, two polynomials p and q, that are equal µ−a.s. (“a.s.” means “almostsurely”), are considered to be the same, for all n ≥ 0. Since µ has finite momentsof any order, we have:

C = F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ L2(Rd, µ).

For all n ≥ 0, Fn is a closed subspace of L2(Rd, µ), since Fn is a finite dimensionalvector space. Let G0 := F0 = C and, for all n ≥ 1, let Gn := Fn ª Fn−1, i.e.,Gn is the orthogonal complement of Fn−1 into Fn. This orthogonal complementis computed with respect to the inner product 〈f, g〉 :=

∫Rd f(x)g(x)µ(dx), for f ,

g ∈ L2(Rd, µ). We define now the Hilbert space

H := ⊕∞n=0Gn ⊂ L2(Rd, µ).

The Hilbert space H can be understood in two ways: either as the orthogonalsum of the countable family of finite dimensional Hilbert spaces Gnn≥0 or asthe closure of the space F , of all polynomial functions of d real variables, withcomplex coefficients, in the space L2(Rd, µ). We would like to mention againthat, in F , two polynomial functions that are equal µ−a.s., are considered to beidentical. We also define F−1 := 0 and G−1 := 0, where 0 denotes the nullspace.

For any i ∈ 1, 2, . . . , d, we denote the multiplication operator by the variablexi, by Xi. The domain of this operator is considered to be the space F describedabove. Thus, if p(x1, x2, . . . , xd) is a polynomial function, we have

Xip(x1, x2, . . . , xd) = xip(x1, x2, . . . , xd). (2.1)

AN INTERACTING FOCK SPACE CHARACTERIZATION 87

We can see that, for any i ∈ 1, 2, . . . , d, Xi maps F into F , and since F is densein H, Xi is a densely defined linear operator on the Hilbert space H. Let us alsoobserve that Xi maps Fn into Fn+1, for all 1 ≤ i ≤ d and n ≥ 0.

If f , g ∈ L2(Rd, µ), such that 〈f , g〉 = 0, we say that f and g are orthogonaland denote this fact by f⊥g.

For all n ≥ 0, let Pn denote the orthogonal projection of H onto Gn. If k andn are two non–negative integers such that k ≥ n + 2, then since Pn maps H ontoGn, Gn ⊂ Fn, and Xi maps Fn into Fn+1, we can see that XiPn maps H intoFn+1. Since n+1 < k, we have Gk⊥Fn+1, and because Pk projects all polynomialfunctions into Gk, we conclude that:

PkXiPn = 0, (2.2)

for all 1 ≤ i ≤ d and all k ≥ n+2. Taking the adjoint in both sides of the equality(2.2), we obtain:

PnXiPk = 0, (2.3)

for all 1 ≤ i ≤ d and all k ≥ n + 2. Thus, we conclude that, for all r and snon–negative integers, such that |r − s| ≥ 2, and for all 1 ≤ i ≤ d, we have:

PrXiPs = 0. (2.4)

Let I be the identity operator of H. Since I =∑

n≥0 Pn, it follows from (2.4)that, for all 1 ≤ i ≤ d,

Xi = IXiI

=

( ∞∑

k=0

Pk

)Xi

( ∞∑n=0

Pn

)

=∑

|k−n|≤1

PkXiPn

=∞∑

n=1

Pn−1XiPn +∞∑

n=0

PnXiPn +∞∑

n=0

Pn+1XiPn. (2.5)

For all i ∈ 1, 2, . . . , d, we define the following three operators:

a−(i) =∞∑

n=1

Pn−1XiPn, (2.6)

a0(i) =∞∑

n=0

PnXiPn, (2.7)

and

a+(i) =∞∑

n=0

Pn+1XiPn. (2.8)

Let us observe that, for any n ≥ 0, the restrictions of these three operators to thespace Gn, are:

a−(i)|Gn = Pn−1XiPn : Gn → Gn−1, (2.9)


a0(i)|Gn = PnXiPn : Gn → Gn, (2.10)

and

a+(i)|Gn = Pn+1XiPn : Gn → Gn+1. (2.11)

We call a−(i), a0(i), and a+(i) the annihilation, preservation (neutral), and cre-ation operators of index i, respectively. We can now rewrite the formula (2.5)as:

Xi = a−(i) + a0(i) + a+(i), (2.12)

for all i ∈ 1, 2, . . . , d. The domain of the operators Xi, a−(i), a0(i), and a+(i),involved in formula (2.12), is considered to be the space F .

For any two linear operators A and B densely defined on the same Hilbert spaceH, we define their commutator [A, B], as:

[A,B] := AB −BA.

It is clear that, if K is a subspace of H, such that K is contained in both domainsof A and B, AK ⊂ K, and BK ⊂ K, then K is contained in the domain of thecommutator [A, B].

Since Fn = G0 ⊕ G1 ⊕ · · · ⊕ Gn, using (2.9), (2.10), and (2.11), we concludethat the space Fn is invariant under the action of the operators a0(i) and [a−(j),a+(k)], i.e., a0(i)Fn ⊂ Fn and [a−(j), a+(k)]Fn ⊂ Fn, for all n ≥ 0 and all i, j,k ∈ 1, 2, . . . , d. We denote by a0(i)|Fn and [a−(j), a+(k)]|Fn the restrictions ofthese operators to the finite dimensional space Fn.

3. Probability measures with square summable support

In this section, we will present the main result of this paper.

Definition 3.1. A probability measure µ on Rd is said to have a square summablesupport if

µ =∞∑

n=1

pnδx(n) , (3.1)

for some sequence pnn≥1, of non–negative real numbers, such that∞∑

n=1

pn = 1,

and some sequence x(n)n≥1, of vectors in Rd, such that∞∑

n=1

∣∣x(n)∣∣2 < ∞, (3.2)

where | · | denotes the Euclidean norm of Rd and δx the Dirac delta measure at x,for any point x in Rd.

The following lemma will be useful in proving the main result of the paper.


Lemma 3.2. For any i ∈ 1, 2, . . . , d, and any n ≥ 0,

Tr([a−(i), a+(i)]|Fn) = ‖ a+(i)|Gn

‖2HS = ‖ a−(i)|Gn+1 ‖2HS , (3.3)

where Tr([a−(i), a+(i)]|Fn) denotes the trace of the restriction of [a−(i), a+(i)]

to the space Fn, and ‖ a+(i)|Gn ‖HS and ‖ a−(i)|Gn+1 ‖HS the Hilbert–Schmidtnorms of the restrictions of a+(i) to Gn and a−(i) to Gn+1, respectively.

Proof. Let i ∈ 1, 2, . . . , d and n ≥ 0 be fixed. For all k ≥ 0, let e(k)u 1≤u≤rk

,be an orthonormal basis of Gk. For all 1 ≤ u ≤ rk, since e

(k)u ∈ Gk, we have:

a+(i)e(k)u = Pk+1Xie

(k)u

=rk+1∑v=1

〈Xie(k)u , e(k+1)

v 〉e(k+1)v

and

a−(i)e(k)u = Pk−1Xie

(k)u

=rk−1∑w=1

〈Xie(k)u , e(k−1)

w 〉e(k−1)w .

Thus, for all k ≥ 0, we have:rk∑

u=1

〈[a−(i), a+(i)]e(k)u , e(k)

u 〉

=rk∑

u=1

〈a−(i)a+(i)e(k)u , e(k)

u 〉 −rk∑

u=1

〈a+(i)a−(i)e(k)u , e(k)

u 〉

=rk∑

u=1

rk+1∑v=1

〈Xie(k)u , e(k+1)

v 〉〈a−(i)e(k+1)v , e(k)

u 〉

−rk∑

u=1

rk−1∑w=1


w 〉〈a+(i)e(k−1)w , e(k)

u 〉

=rk∑

u=1

rk+1∑v=1

〈Xie(k)u , e(k+1)

v 〉〈Xie(k+1)v , e(k)

u 〉

−rk∑

u=1

rk−1∑w=1


w 〉〈Xie(k−1)w , e(k)

u 〉

=rk∑

u=1

rk+1∑v=1

〈Xie(k)u , e(k+1)

v 〉〈e(k+1)v , Xie

(k)u 〉

−rk∑

u=1

rk−1∑w=1

〈e(k)u , Xie

(k−1)w 〉〈Xie

(k−1)w , e(k)

u 〉

=rk∑

u=1

rk+1∑v=1

| 〈Xie(k)u , e(k+1)

v 〉 |2 −rk−1∑w=1

rk∑u=1

| 〈Xie(k−1)w , e(k)

u 〉 |2 . (3.4)


Summing in formula (3.4), from k = 0 to k = n, and using the fact that, for k = 0,∑rk−1w=1

∑rk

u=1 | 〈Xie(k−1)w , e

(k)u 〉 |2= 0 (since G−1 = 0), we obtain:

Tr([a−(i), a+(i)]|Fn) =

n∑

k=0

rk∑u=1

〈[a−(i), a+(i)]e(k)u , e(k)

u 〉

=rn∑

u=1

rn+1∑v=1

| 〈Xie(n)u , e(n+1)

v 〉 |2 (3.5)

=rn∑

u=1

rn+1∑v=1

| 〈a+(i)e(n)u , e(n+1)

v 〉 |2

=rn∑

u=1

‖ a+(i)e(n)u ‖2

= ‖ a+(i)|Gn ‖2HS .

It follows also from (3.5) that:

Tr([a−(i), a+(i)]|Fn) =rn∑

u=1

rn+1∑v=1

| 〈Xie(n)u , e(n+1)

v 〉 |2

=rn+1∑v=1

rn∑u=1

| 〈Xie(n+1)v , e(n)

u 〉 |2

=rn+1∑v=1

rn∑u=1

| 〈a−(i)e(n+1)v , e(n)

u 〉 |2

=rn+1∑v=1

‖ a−(i)e(n+1)v ‖2

= ‖ a−(i)|Gn+1 ‖2HS .

Hence the lemma is proved. ¤

The following theorem characterizes the probability measures, with a squaresummable support, in terms of their preservation and commutators between theannihilation and creation operators.

Theorem 3.3. A probability measure µ on Rd has a square summable support ifand only if it has finite moments of any order and, for all i ∈ 1, 2, . . . , d, thesequence

Tr

((a0(i)|Fn)2

)n≥0

is bounded and

∞∑n=0

Tr([a−(i), a+(i)]|Fn) < ∞. (3.6)

Proof. Part 1: NecessityLet us assume that µ has a square summable support. Then

µ =∞∑

n=1

pnδx(n) ,


with∑∞

n=1

∣∣x(n)∣∣2 < ∞.

Let R2 :=∑∞

n=1

∣∣x(n)∣∣2 < ∞. It is clear that µ is a discrete measure with

compact support contained in the ball B[0, R] := x ∈ Rd | |x| ≤ R. Since µ hascompact support, it has finite moments of any order. From the compactness ofthe support of µ it also follows that the space F , of all polynomial functions of dvariables: x1, x2, . . . , xd, is dense in L2(Rd, µ). Thus H = ⊕∞n=0Gn = L2(Rd, µ).Moreover, for all i ∈ 1, 2, . . . , d, the operator Xi, of multiplication by the variablexi, is a bounded operator from L2(Rd, µ) to L2(Rd, µ).

Since µ =∑∞

n=1 pnδx(n) , enn≥1 is an orthonormal basis for L2(Rd, µ), whereen := 1√

pn1x(n), for all n ≥ 1, such that pn > 0 (it is possible that the measure

µ has a finite support, in which case, all the pn’s are zero, except finitely manyof them). For all n ≥ 1 and i ∈ 1, 2, . . . , d, we denote the ith component of thevector x(n) by x

(n)i . We also denote the norm of the space L2(Rd, µ) by ‖ · ‖.

For all i ∈ 1, 2, . . . , d, we have:

‖Xi‖2HS =∑

n≥1

‖Xien‖2

=∑

n≥1

‖x(n)i en‖2

=∑

n≥1

(x

(n)i

)2‖en‖2

=∑

n≥1

(x

(n)i

)2

≤∑

n≥1

∣∣x(n)∣∣2

= R2

< ∞.

Thus Xi is a Hilbert–Schmidt operator, for all i ∈ 1, 2, . . . , d.For each n ≥ 0, let e(n)

u 1≤u≤rn be an orthonormal basis for Gn. Let

U =e(0)u

1≤u≤r0

⋃ e(1)u

1≤u≤r1

⋃ e(2)u

1≤u≤r2

⋃· · · .

Then U is an orthonormal basis for H.Using now the fact that the multiplication operator Xi is the sum of the creation,

preservation, and annihilation operators of index i, we conclude that, for all i ∈ 1,2, . . . , d, we have:

‖Xi‖2HS

=∞∑

n=0

rn∑u=1

‖Xie(n)u ‖2

=∞∑

n=0

rn∑u=1

‖a+(i)e(n)u + a0(i)e(n)

u + a−(i)e(n)u ‖2.


Since, for any n ≥ 0, and any u ∈ 1, 2, . . . , rn, a+(i)e(n)u ∈ Gn+1, a0(i)e(n)

u ∈ Gn,a−(i)e(n)

u ∈ Gn−1, and the spaces Gn+1, Gn, and Gn−1 are orthogonal, we have:

‖Xi‖2HS =∞∑

n=0

rn∑u=1

(‖a+(i)e(n)

u ‖2 + ‖a0(i)e(n)u ‖2 + ‖a−(i)e(n)

u ‖2)

=∞∑

n=0

rn∑u=1

‖a+(i)e(n)u ‖2 +

∞∑n=0

rn∑u=1

‖a0(i)e(n)u ‖2 +

∞∑n=0

rn∑u=1

‖a−(i)e(n)u ‖2

= ‖a+(i)‖2HS + ‖a0(i)‖2HS + ‖a−(i)‖2HS .

Because ‖Xi‖HS < ∞, we get ‖a+(i)‖HS < ∞ and ‖a0(i)‖HS < ∞, for alli ∈ 1, 2, . . . , d.

Now, let us observe that a0(i)|Fn is self–adjoint with respect to inner product〈·, ·〉 of the space L2(Rd, µ), for all n ≥ 0. Therefore, we can see that:

‖a0(i)‖2HS = supn≥0

‖a0(i)|Fn‖2HS

= supn≥0

n∑

k=0

rk∑u=1

〈a0(i)e(k)u , a0(i)e(k)

u 〉

= supn≥0

n∑

k=0

rk∑u=1

⟨(a0(i)

)2e(k)u , e(k)

u

⟩

= supn≥0

Tr((a0(i)|Fn)2

).

This implies that the sequenceTr

((a0(i)|Fn)2

)n≥0

is bounded, for all i ∈1, 2, . . . , d.

On the other hand, from Lemma 3.2, we know that,

‖a+(i)|Gn‖2HS = Tr([a−(i), a+(i)]|Fn),

for all i ∈ 1, 2, . . . , d. Thus

‖a+(i)‖2HS =∞∑

n=0

‖a+(i)|Gn‖2HS

=∞∑

n=0

Tr([a−(i), a+(i)]|Fn).

Hence∑∞

n=0 Tr([a−(i), a+(i)]|Fn) < ∞, for all i ∈ 1, 2, . . . , d.

Part 2: SufficiencyLet us suppose that µ is a probability measure on Rd, with finite moments of

any order, such that, for all i ∈ 1, 2, . . . , d,∞∑

n=0

Tr([a−(i), a+(i)]|Fn) < ∞

and the sequenceTr

((a0(i)|Fn)2

)n≥0

is bounded.


We have seen before that

‖a+(i)‖2HS =∞∑

n=0

Tr([a−(i), a+(i)]|Fn).

It also follows from Lemma 3.2, that

‖a−(i)‖2HS =∞∑

n=0

Tr([a−(i), a+(i)]|Fn).

Thus a+(i) and a−(i) are Hilbert–Schmidt operators from the Hilbert space H toitself, for all i ∈ 1, 2, . . . , d.

We have also seen before that the fact that the sequenceTr

((a0(i)|Fn)2

)n≥0

is bounded is equivalent to the fact that a0(i) is a Hilbert–Schmidt operator fromH to H. Thus, it follows, as before, that

‖Xi‖2HS = ‖a+(i)‖2HS + ‖a0(i)‖2HS + ‖a−(i)‖2HS < ∞.

Hence the multiplication operator Xi is a Hilbert–Schmidt operator from H to H,for all i ∈ 1, 2, . . . , d. Being a Hilbert–Schmidt operator, Xi is also a boundedoperator on H, for all i ∈ 1, 2, . . . , d. Let Ri := ‖Xi‖H,H be the operatornorm of Xi on H. Hence for any polynomial function g of d variables, we have‖Xig‖ ≤ Ri‖g‖. We denote by E[·] the expectation with respect to µ.

Let ε > 0 be fixed and let Bi = (x1, x2, . . . , xd) ∈ Rd | |xi| ≥ Ri + ε. Thenfor all n ≥ 1,

(Ri + ε)2nµ(Bi) ≤ E[x2ni 1Bi ]

≤ E[x2ni 1]

= ‖Xni 1‖2

≤(‖Xn

i ‖H,H · ‖1‖)2

≤(‖Xi‖n

H,H · 1)2

= R2ni .

Thus we obtain µ(Bi) ≤ R2ni /(Ri + ε)2n, for all n ≥ 1, and letting n → ∞, we

conclude that µ(Bi) = 0, for all ε > 0. Hence the support of the probabilitymeasure µ is contained in the set

Ci := (x1, x2, . . . , xd) ∈ Rd | |xi| ≤ Ri,for all i ∈ 1, 2, . . . , d. Therefore, µ has compact support contained in the set∩d

i=1Ci. Since µ has compact support, the space F of all polynomial functionsis dense in L2(Rd, µ) and thus H = L2(Rd, µ). Therefore, Xi is Hilbert–Schmidtand, in particular, bounded from L2(Rd, µ) into L2(Rd, µ). The operator Xi isalso self–adjoint on L2(Rd, µ), for all i ∈ 1, 2, . . . , d.


From the general form of the self–adjoint Hilbert–Schmidt operators on a Hilbertspace, we know that the spectrum of Xi is discrete and coincides with the pointspectrum. That means, for all i ∈ 1, 2, . . . , d, there exist a sequence of realnumbers λ(i)

n n≥1 and an orthonormal basis f (i)n n≥1 of L2(Rd, µ), such that,

for all h ∈ L2(Rd, µ),

Xih =∞∑

n=1

λ(i)n 〈h, f (i)

n 〉f (i)n . (3.7)

Moreover,∞∑

n=1

(λ(i)

n

)2 = ‖Xi‖2HS < ∞. (3.8)

For all n ≥ 1, we have Xif(i)n = λ

(i)n f

(i)n . This means

(xi− λ

(i)n

)f

(i)n = 0, µ–a.s..

Since ‖f (i)n ‖ = 1, we know that f

(i)n cannot be equal to zero µ–a.s.. Thus the

hyperplane π(i)n := (x1, x2, . . . , xn) ∈ Rd | xi = λ

(n)i has a positive probability,

i.e., µ(π(i)n ) > 0. On the complement of this hyperplane f

(i)n (x) = 0, µ–a.s..

This means that f(i)n 1(

π(i)n

)c = 0, µ–a.s., where 1(π

(i)n

)c denotes the characteristic

function of the complement of π(i)n . Let g

(i)n := f

(i)n 1

π(i)n

. Then

f (i)n = f (i)

n 1π

(i)n

+ f (i)n 1(

π(i)n

)c

= g(i)n + 0

= g(i)n , µ− a.s..

Thus, we can replace the orthonormal basis f (i)n by g(i)

n , in Equation (3.7), toobtain the equality:

Xih =∞∑

n=1

λ(i)n 〈h, g(i)

n 〉g(i)n ,

for all h ∈ L2(Rd, µ), where, for all n ≥ 1, the support of g(i)n is contained in the

hyperplane π(i)n . Since g(i)

n n≥1 is an orthonormal basis of L2(Rd, µ), we have:

µ

([ ∞⋃n=1

π(i)n

]c)= ‖1[

∪∞n=1π(i)n

]c‖2

=∞∑

n=1

⟨1[∪∞n=1π

(i)n

]c , g(i)n

⟩2

= 0.

Hence, for all i ∈ 1, 2, . . . , d, the support of µ is contained in the union of thehyperplanes π

(i)n , for n ≥ 1.


If λ is an eigenvalue of Xi, and λ 6= 0, then the eigenspace corresponding to λ

is finite dimensional, because of the condition∑∞

n=1

(λ

(i)n

)2< ∞. That means, if

λ 6= 0, then the set n ∈ N | λ(i)n = λ is finite.

Let i ∈ 1, 2, . . . , d and λ = λ(i)n 6= 0, for some n ≥ 0, be fixed. If k denotes

the multiplicity of λ, as an eigenvalue of Xi, we conclude that, for any sequenceBll≥1, of disjoint Borel subsets of the hyperplane π := (x1, x2, . . . , xd) | xi =λ, there are at most k sets Bl1 , Bl2 , . . . , such that µ(Bl1) > 0, µ(Bl2) > 0,. . . . This is true, since the characteristic functions 1Bl1

, 1Bl2, . . . are non–zero

orthogonal eigenvectors of the multiplication operator Xi, corresponding to thesame eigenvalue λ.

For all n ∈ N, let Cn be the family of cubes, of π, of the form

Kn,r = π ∩

(x1, . . . , xd) ∈ Rd | r1

2n≤ x1 <

r1 + 12n

, . . . ,rd

2n≤ xd <

rd + 12n

,

where r = (r1, . . . , rd) ∈ Zd. It is clear that for all r 6= s, Kn,r ∩Kn,s = ∅. SinceKn,rr∈Zd is a partition of π composed of mutually disjoint Borel subsets, weconclude that at most k of the sets Kn,rr∈Zd have a positive probability measureµ. For all n ∈ N, let tn be the cardinality of the set An := r ∈ Zd | µ(Kn,r) > 0.Then, for each n ∈ N, tn is a natural number less than or equal to k. Let usobserve that, since each cube Kn,r, from Cn, can be written as a finite unionof cubes Kn+1,s, from Cn+1, for each r ∈ An, there exists at least one cubeKn+1,sr ∈ Cn+1, such that Kn+1,sr ⊂ Kn,r and µ(Kn+1,sr ) > 0. Thus sr ∈ An+1.For each r ∈ An, we choose one sr and fix it. If r1, r2 ∈ An, such that r1 6= r2,we have Kn,r1 ∩ Kn,r2 = ∅, and since Kn+1,sr1

⊂ Kn,r1 and Kn+1,sr2⊂ Kn,r2 ,

we conclude that Kn+1,sr1∩Kn+1,sr2

= ∅. Thus sr1 6= sr2 and so, the mappingr 7→ sr is a one–to–one function from An to An+1. Hence the cardinality of An

does not exceed the cardinality of An+1, or equivalently tn ≤ tn+1, for all n ∈ N.Therefore, t1 ≤ t2 ≤ t3 ≤ · · · ≤ k. Since tnn≥1 is a bounded non–decreasingsequence of natural numbers, we conclude that it must be stationary, i.e., thereexists n0 ∈ N, such that tn0 = tn0+1 = tn0+2 = · · · . From the fact that, foreach n ≥ n0, tn = tn+1, it follows that, for each r ∈ An, there exists a uniquesr ∈ An+1, such that Kn+1,sr ⊂ Kn,r. This uniqueness property implies thatµ(Kn+1,sr ) = µ(Kn,r). Let An0 = r1, r2, . . . , rtn0

. For any j ∈ 1, 2, . . . , tn0,we can construct a decreasing sequence of cubes K(n)

j n≥n0 , in the following way:

K(n0)j := Kn0,rj , K

(n0+1)j is the unique cube from Cn0+1, that is contained is Kn0,rj

and has a positive probability measure µ, K(n0+2)j is the unique cube from Cn0+2

that is contained in K(n0+1)j and has a positive probability measure µ, and so on.

Thus, we obtain a decreasing sequence of cubes: K(n0)j ⊃ K

(n0+1)j ⊃ K

(n0+2)j ⊃ · · ·

such that µ(K(n0)j ) = µ(K(n0+1)

j ) = µ(K(n0+2)j ) = · · · > 0. Since the diameter

of the cube K(n)j (i.e., the supremum of the distances between any two points of

the cube) tends to 0, as n →∞, we know that the intersection of all these cubesis either the empty set or a set that contains only one point. By the monotoneconvergence theorem, we have: µ(∩n≥n0K

(n)j ) = limn→∞ µ(K(n)

j ) = µ(K(n0)j ) > 0.


Thus ∩n≥n0K(n)j 6= ∅. Consequently, for all j ∈ 1, 2, . . . , tn0, there exists

x(j) ∈ π, such that ∩n≥n0K(n)j = x(j) and µ(x(j)) = µ(K(n0)

j ) > 0. Hence, wehave:

µ(π) = µ(∪r∈ZdKn0,r)

=tn0∑

j=1

µ(K(n0)j )

=tn0∑

j=1

µ(x(j))

= µ(x(1), x(2), . . . , x(tn0 )).This implies that the restriction of the probability measure µ to the Borel subsetsof the hyperplane π is a finite combination of Dirac delta measures. Therefore, foreach λ

(i)n 6= 0, there exist finitely many points y

(i)1,n, y

(i)2,n, . . . , y

(i)sn,i,n in π

(i)n , such

that, for any Borel subset C of π(i)n ,

µ(C) =sn,i∑u=1

p(i)u,nδ

y(i)u,n

(C), (3.9)

where p(i)u,n := µ(y(i)

u,n) > 0, for all u ∈ 1, 2, . . . , sn,i. The number of thesepoints, sn,i, coincides with the multiplicity of the eigenvalue λ

(i)n . Hence

‖Xi‖2HS =∞∑

n=1

(λ(i)

n

)2

= ξi,

where ξi denotes the sum of the squares of the ith coordinates of y(i)u,n, for n ≥ 1

and 1 ≤ u ≤ sn,i. The only eigenvalue of Xi that might have an infinite dimen-sional eigenspace is λ = 0, eventually. Thus, at this moment, we do not knowthe behavior of the probability measure µ on the Borel subsets of the hyperplane(x1, x2, . . . , xd) ∈ Rd | xi = 0. We may call such a hyperplane a “bad” hyper-plane. We should not forget though, that our conclusion, regarding the fact that µis a finite combination of delta measures, on each hyperplane of equation xi = λ,for λ 6= 0, is true for all i ∈ 1, 2, . . . , d. This means that we know the behavior ofµ everywhere, except on the intersection of all the bad hyperplanes. Fortunately,we have

d⋂

i=1

(x1, x2, . . . , xd) ∈ Rd

∣∣ xi = 0

= (0, 0, . . . , 0).

Hence besides the set

D :=d⋃

i=1

∞⋃n=1

sn,i⋃u=1

y(i)u,n, (3.10)

the support of µ might contain eventually only one more point, namely 0, the zerovector of Rd. There are many repetitions among the singleton sets y(i)

u,n, that


participate in the unions from the right–hand side of (3.10). For example, if apoint y

(i)u,n has all the coordinates different from zero, then 1y(i)

u,n is a non–zeroeigenvector, corresponding to a non–zero eigenvalue, for each of the multiplicationoperators Xj , 1 ≤ j ≤ d. However, if a point y

(i)u,n is different from all the points

y(j)v,m, for a fixed j and all values of m and v, then the jth coordinate of y

(i)u,n is zero.

Thus, when we compute the sum of the squares of the jth coordinates of all thepoints from the support of µ, the point y

(i)u,n does not contribute with anything.

This fact is very important in proving the square summability of the support ofµ. Let us rewrite the set ∪d

i=1 ∪∞n=1 ∪sn,i

u=1y(i)u,n as x(n)N

n=1, where x(k) 6= x(l),for all k 6= l, and N could be a finite positive integer or ∞. Then,

µ = p0δ0 +N∑

n=1

pnδx(n) , (3.11)

where, for all n ≥ 0, pn ≥ 0 (if 0 is not in the spectrum of µ, then p0 = 0), and∑Nn=0 pn = 1. Thus, we have:

N∑n=1

|x(n)|2 =d∑

i=1

N∑n=1

|x(n)i |2

=d∑

i=1

ξi

=d∑

i=1

‖Xi‖2HS

< ∞.

This proves that µ has a square summable support. ¤If d = 1 and Vn denotes the space of all polynomial functions, of one real

variable, with complex coefficients, of degree at most n, then, since the algebraiccodimension Vn into Vn+1 is 1, we conclude that the dimension of Gn is at most 1,for all n ≥ 0. In fact the dimension of Gn is equal to 1, for all n ≥ 0, if and onlyif the support of the measure µ is an infinite set, in which case Fn = Vn, for alln ≥ 0 (we should remember that Fn is the space Vn factorized to the equivalencerelation given by the µ−almost sure equality). In that case, since the dimension ofGn is 1, there exists a unique polynomial fn in Gn that has the leading coefficientequal to 1, for all n ≥ 0. Since we have only one multiplication operator X1, oneannihilation operator a+(1), one preservation operator a0(1), and one annihilationoperator a−(1), we can denote them simply by X, a+, a0, and a−, respectively.Also, sice fn ∈ Gn and a− : Gn → Gn−1, there exists a unique real number ωn,such that a−fn = ωnfn−1, for all n ≥ 1 (for n = 0, sice G−1 = 0, we candefine ω0 := 0 and f−1 := 0). Similarly, there exists a unique real number αn,such that a0fn = αnfn, for all n ≥ 0. Since both fn+1 and Xfn have the leadingcoefficient equal to 1, we conclude that a+fn = fn+1, for all n ≥ 0. Thus, sinceX = a+ + a0 + a−, we obtain that, for all n ≥ 0,

Xfn = fn+1 + αnfn + ωnfn−1. (3.12)


The sequences αnn≥0 and ωnn≥1, are called the Szego–Jacobi parameters ofµ. It is easy to see that [a−, a+]fk = (ωk+1 − ωk)fk, for all k ≥ 0, and thussince ω0 = 0, if one considers the algebraic base fk0≤k≤n (or the normalizedorthogonal base (1/ ‖ fk ‖)fk0≤k≤n) of Fn, then

Tr([a−, a+]|Fn) =n∑

k=0

(ωk+1 − ωk)

= ωn+1,

for all n ≥ 0. Similarly, since (a0)2fk = α2kfk, for all k ≥ 0, we conclude that

Tr((a0|Fn)2) =

n∑

k=0

α2k,

for all n ≥ 0. If the support of µ is a finite set, then we can still make sense ofthe formula (3.12), by defining fn := 0, αn := 0, and ωn := 0, for n large enough.Thus, from Theorem 3.3, we obtain the following corollary:

Corollary 3.4. Let µ be a probability measure on R having finite moments ofany order. Then µ has a square summable support if and only if both series∑∞

n=0 α2n and

∑∞n=1 ωn are convergent, where αnn≥0 and ωnn≥1 denote the

Szego–Jacobi parameters of µ.

Acknowledgements. Part of this work was done during the visit of H.–H. Kuoto the Vito Volterra Center from April 1 to July 31, 2005. He is grateful for thefinancial support, from the “Commissione Per Gli Scambi Culturali Fra L’Italia egli Stati Uniti” (Italian Fulbright Commission), for a Fulbright Lecturing grant.

References

1. Accardi, L. and Bozejko, M.: Interacting Fock space and Gaussianization of probabilitymeasures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998) 663–670.

2. Accardi, L., Kuo, H.-H., and Stan, A. I.: Characterization of probability measures throughthe canonically associated interacting Fock spaces, Infin. Dimens. Anal. Quantum Probab.Relat. Top. 7 No. 1 (2005) 485–505.

3. Accardi, L., Kuo, H.-H., and Stan, A. I.: Moments and commutators of probability measures,Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 No. 4 (2007) 591–612.

4. Accardi, L., Lu, Y. G., and Volovich, I.: The QED Hilbert module and interacting Fockspaces, IIAS Reports No. 1997–008 (1997) International Institute for Advanced Studies,Kyoto.

5. Accardi, L. and Nahni, M.: Interacting Fock space and orthogonal polynomials in severalvariables, Preprint (2002).

6. Accardi, L. and Skeide, M.: Interacting Fock space versus full Fock module, Volterra PreprintNo. 328 (1998).

7. Parthasarathy, K. R.: An Introduction to Quantum Stochastic Calculus, Birkhauser, 1992.8. Shohat, J. A. and Tamarkin, J. D.: The Problem of Moments, Mathematical Surveys, Num-

ber I, American Mathematical Society, 1943.9. Stieltjes, T. J.: Recherches sur les fractiones continues, Annales de la Faculte des Sciences

de Toulouse (1) 8 (1894), T 1–122, (1) 9 (1895) A5–47.10. Szego, M.: Orthogonal Polynomials, Coll. Publ. 23, Amer. Math. Soc., 1975.


Luigi Accardi: Centro Vito Volterra, Facolta di Economia, Universita di RomaTor Vergata, 00133 Roma, Italy


Hui-Hsiung Kuo: Department of Mathematics, Louisiana State University, BatonRouge, LA 70803, U.S.A.


Aurel I. Stan: Department of Mathematics, The Ohio State University, 1465 MountVernon Avenue, Marion, OH 43302, U.S.A.


OPTIMAL CONSUMPTION AND PORTFOLIO FOR ANINSIDER IN A MARKET WITH JUMPS

DELPHINE DAVID AND YELIZ YOLCU OKUR

Abstract. We examine a stochastic optimal control problem in a financialmarket driven by a Levy process with filtration F = (Ft)t∈[0,T ]. We assumethat in the market there are two kinds of investors with different levels ofinformation: an uninformed agent whose information coincides with the nat-ural filtration of the price processes and an insider who has more informationthan the uninformed agent. When optimal consumption and investment ex-ist, we identify some necessary conditions and find the optimal strategy byusing forward integral techniques. We conclude by giving some examples.

1. Introduction

The consumption-portfolio problem in continuous time market models was firstintroduced by Merton [22], [23]. He worked with the assumption that stock priceswere governed by Markovian dynamics with constant coefficients. This approachis based on stochastic dynamic programming. For a market which consists of onlytwo assets, he formulated the problem of choosing optimal portfolio selection andconsumption rules as follows:

max(c,X)

E

[∫ T

0

U(c(t), t)dt+ g(X(T ), T )

],

subject to the budget constraint, c(t) ≥ 0, X(t) > 0 for all t ∈ [0, T ] withX(0) = x.Here X(·) represents the wealth process, c(·) is the consumption per unit time, U isassumed to be a strictly concave utility function and g is the bequest valuation func-tion which is concave in terminal wealth X(T ). Recently, many authors have useda martingale representation technique instead of dynamic programming methods:see Cox and Huang [4], [5], Karatzas, Lehoczky and Shreve [15] and Pliska [26].In incomplete markets, the theory was studied by He and Pearson [12], Karatzaset al. [16], Karatzas and Zitkovic [18], Kramkov and Schachermayer [20].

In financial markets a trader is assumed to make her decisions with respect tothe information revealed by the market events. This information is assumed to beaccessible to everyone. However, in reality some agents have superior information

2000 Mathematics Subject Classification. Primary 93E20 91B28; Secondary 60H05 60G57.Key words and phrases. Insider trading, forward integral, Levy process, portfolio optimiza-

tion.* This research was partially supported by grants from AMaMeF (Projects No #2062 and

#1850).

101



102 DELPHINE DAVID AND YELIZ YOLCU OKUR

about the market. The informed agent possesses information regarding some fu-ture movements in stock prices and she has opportunities to gain profits by tradingbefore prices reach equilibrium. Because of this fact, it is more realistic to modelstochastic control problems where the information is non-identical. This paperfocuses on the characterization of the optimal consumption and portfolio choicesof an insider when the market is driven by a Levy process. We also compare theperformance functions of informed and uninformed agents in certain specific cases.These results can be helpful in detecting informed agents.

Karatzas and Pikovsky [17]’s study on stochastic control problems with thepresence of insiders is one of the earliest studies regarding to this approach. In thisstudy, they assume that the informed agents maximize their expected logarithmicutility from terminal wealth and consumption in Brownian motion framework.See also Amendinger et al. [1], Grorud et al. [11] and Elliott et al. [8]. In asubsequent, Grorud [10] analyzed the optimal portfolio and consumption choiceswhen prices are driven by a Brownian motion and a compound Poisson process.The first general study of insider trading based on forward integrals (withoutassuming the enlargement of filtration) was done in Biagini and Øksendal [3].Thereafter, many authors used Malliavin calculus and forward integration to studythe optimal portfolio choices of insiders. See e.g. Biagini and Øksendal [2], Imkeller[13], Kohatsu-Higa and Sulem [19] and Leon et al. [21] for further discussions ofthe Brownian motion case. An extension of forward integration in to the case ofcompensated Poisson random measures was proposed by Di Nunno et al. [6]. Thatsetting is used in solving the optimal portfolio problem as elaborated by Di Nunnoet al. [7] and in solving the optimal consumption rate problem as elaborated byØksendal [25].

In this paper, we extend the results of Di Nunno et al. [7] and of Øksendal[25] by considering both the optimal portfolio and consumption rate choices of aninsider when her portfolio is allowed to anticipate the future. We formulate theassociated optimal control problem as follows:

max(c,π)

E

[∫ T

0

e−δ(t) ln c(t)dt+Ke−δ(T ) lnX(c,π)(T )

].

δ(t) ≥ 0 is a given bounded deterministic function representing the discount rate,K is a nonnegative constant representing the weight of the expected utility of theterminal wealth in the performance function, T > 0 is a fixed terminal time andX(c,π)(·) is the wealth process with control parameters (c, π). As we consider theoptimal consumption and the optimal portfolio problems together, the solution forthe optimal portfolio is more complex than the studies which consider only theoptimization with respect to terminal wealth. Moreover, unlike many studies wedo not use the initial enlargement of filtration technique. As a result, we provethat if there exist optimal portfolio and consumption, then the F-Brownian motionB(·) is a G = (Gt)t∈[0,T ]-semimartingale where Ft ⊂ Gt for all t ∈ [0, T ] and weshow that it also holds for Levy processes (see Theorem 4.5).

This paper is organized as follows. In Section 2, we recall some mathematicalpreliminaries about forward integrals which are relevant to our calculations. In

OPTIMAL CONSUMPTION AND PORTFOLIO FOR AN INSIDER 103

Section 3, we introduce the main problem. In Section 4, we characterize the opti-mal consumption and portfolio choice for the problem introduced in the previoussection. In Section 5, we compare the optimal wealth process and the performancefunction of informed and uninformed agents in certain specific examples using theresults in Section 4.

2. Preliminary Notes

In this section, we recall the forward integrals with respect to the Brownianmotion and to the compensated Poisson random measure. For further informationon the forward integration with respect to the Brownian motion, we refer to Russoand Vallois [27], [28] and [29], Nualart [24], Biagini and Øksendal [2]. For theforward integration with respect to the compensated Poisson random measure werefer to Di Nunno et al. [6] and [7].

Let (Ω,F , P ) be a product of probability space such that

(Ω, P ) = (ΩB × Ωη, PB ⊗ Pη)

on which are respectively defined a standard Brownian motion B(t)0≤t≤T anda pure jump Levy process, η(t)0≤t≤T , such that

η(t) =∫ t

0

∫

RzN(dt, dz),

where N(dt, dz) = (N − νF )(dt, dz) = N(dt, dz) − νF (dz)dt is a compensatedPoisson random measure with Levy measure νF .

Let B denote the σ-field of Borel sets.

FBt := σB(s), s ≤ t, t ∈ [0, T ] ∨ N

andF N

t := σN(4) : 4 ∈ B(R0 × (0, t)), 0 ≤ t ≤ T ∨ Nare the augmented filtrations generated by B(·) and N(·, ·) respectively. We denoteby Ft = FB

t ⊗ F Nt , 0 ≤ t ≤ T the augmented filtration generated by B and N .

Let G = (Gt)t∈[0,T ] be the filtration such that

Ft ⊂ Gt ⊂ F ∀t ∈ [0, T ],

where T > 0 is a fixed terminal time.Let ϕ(t, ω) be a G-adapted process. Then

∫ T

0

ϕ(t, ω)dB(t) (2.1)

makes no sense in the normal settings. Similarly, for aG-adapted process ψ(t, z, ω),∫ T

0

∫

R0

ψ(t, z, ω)N(dt, dz) (2.2)

does not make sense either. Therefore, it is natural to use forward integrals tohandle this problem and to make the integrals (2.1) and (2.2) well defined.


Definition 2.1. Let ϕ(·, ω), ω ∈ ΩB be a measurable process. The forwardintegral of ϕ with respect to Brownian motion is defined by

∫ ∞

0

ϕ(t, ω)d−B(t) = limε→0

∫ ∞

0

ϕ(t, ω)B(t+ ε)−B(t)

εdt

if the limit exists in probability. Then ϕ is called forward integrable with respectto Brownian motion. If the limit exists also in L2(P ), we write ϕ ∈ DB .

In particular, we recall the following result.

Lemma 2.2. Let ϕ be forward integrable and caglad (i.e., left continuous withright limits). Then for any partition 0 = t0 < t1 < . . . < tN = T

∫ T

0

ϕ(t, ω)d−B(t) = lim|4t|→0

∑

j

ϕ(tj)4B(tj),

where 4B(tj) = B(tj+1)−B(tj) and | 4t |= supj=0,...,N−14tj.Remark 2.3. Let ϕ ∈ DB be a caglad process. If B(·) is a G-semimartingale, then∫ T

0ϕ(t, ω)dB(t) exists as a semimartingale integral and

∫ T

0

ϕ(t, ω)d−B(t) :=∫ T

0

ϕ(t, ω)dB(t).

Let us now give the corresponding definition of forward integral with respect tothe compensated Poisson random measure.

Definition 2.4. Let ϕ(·, z, ω), z ∈ R0, ω ∈ Ωη be a measurable random field. Theforward integral of ϕ with respect to the compensated Poisson random measure isdefined by

∫ ∞

0

∫

R0

ϕ(t, z)N(d−t, dz) = limn→∞

∫ ∞

0

∫

Un

ϕ(t, z) N(dt, dz)

if the limit exists in probability. Here, Un is an increasing sequence of compactsets where Un ⊆ R0, νF (Un) < ∞ and

⋃∞n=1 Un = R0. Then, ϕ is called forward

integrable with respect to the Poisson random measure. If the limit exists in L2(P ),we write ϕ ∈ DN .

Remark 2.5. Let ϕ ∈ DN be caglad. If∫ ·0

∫R0ϕ(t, z, ω)N(dt, dz) is a G-semimartin-

gale, then∫ T

0

∫

R0

ϕ(t, z, ω)N(d−t, dz) :=∫ T

0

∫

R0

ϕ(t, z, ω)N(dt, dz), a.s..

The last result we recall in this section is the Ito formula for forward integrals.We first define the forward process.

Definition 2.6. A forward process is a measurable stochastic function X(·), thatadmits the representation

X(t) = X(0)+∫ t

0

α(s)ds+∫ t

0

β(s)d−B(s)+∫ t

0

∫

R0

γ(s, z)N(d−s, dz), ∀t ∈ [0, T ],


where∫ T

0

(|α(s)|+ β(s)2)ds <∞, almost surely, γ(t, z) is continuous in z around

zero for a.a. t ∈ [0, T ] such that∫ t

0

∫

R0

|γ(s, z)|2 νF (ds, dz) <∞, almost surely for t ∈ [0, T ].

Moreover, β(·) and γ(·, ·) are forward integrable with respect to Brownian motionand compensated Poisson random measure. A shorthand notation for this is

d−X(t) = α(t)dt+ β(t)d−B(t) +∫

R0

γ(t, z)N(d−t, dz). (2.3)

Theorem 2.7. (Ito formula for forward integrals).Let X(·) be a forward process of the form (2.3) and define Y (t) = f(X(t)) for anyf ∈ C2(R) and all t ∈ [0, T ]. Then Y (·) is also a forward process and

d−Y (t) =[f ′(X(t))α(t) +

12f ′′(X(t))β(t)2 +

∫

R0

f(X(t−) + γ(t, z))

− f(X(t−))− f ′(X(t−))γ(t, z)νF (dz)

]dt+ f ′(X(t))β(t)d−B(t)

+∫

R0

(f(X(t−) + γ(t, z))− f(X(t−))

)N(d−t, dz),

where f ′ and f ′′ are the first and second derivatives respectively.

Proof. We refer to Russo and Valois [28] for the proof in the Brownian motioncase and to Di Nunno et al. [6] for the pure jump Levy process case. ¤

3. The Main Problem

Assume there is a riskless and a risky asset in an arbitrage-free financial market.The price per unit of the riskless asset is denoted by S0(·) and satisfies the followingordinary differential equation (O.D.E.)

dS0(t) = r(t)S0(t)dt,S0(0) = 1.

The risky asset has a price process S1(·) defined by

dS1(t) = S1(t−)[µ(t)dt+ σ(t)dB(t) +∫

R0

γ(t, z)N(dt, dz)],

S1(0) > 0.

Assume that the coefficients r(t) = r(t, ω), µ(t) = µ(t, ω), σ(t) = σ(t, ω), andγ(t, z) = γ(t, z, ω) satisfy the following conditions:

(1) r(·), µ(·), σ(·), γ(·, z) are F-adapted caglad processes.(2) γ(t, z) > −1 dt× νF (dz)-a.e.

(3)∫ T

0

|r(t)|+ |µ(t)|+ σ(t)2 +∫

R0

γ(t, z)2 νF (dz)dt <∞ a.s.


In this paper, we will consider an agent who wants to maximize his expected in-tertemporal utility of consumption and terminal value of wealth when the portfoliois allowed to anticipate the future. Hence, we assume that the informed agent’sportfolio and consumption choices are adapted to the larger filtration G. Note thatG = (Gt)t∈[0,T ] can not be chosen freely. For example, if Gt = Ft ∨ σ(X), whereX is any hedgeable FT -measurable random variable, then the informed agent im-mediately obtain the arbitrage opportunity. For further information, see [8], [9],[11] and [17]. In this and the next section, we define the stochastic control prob-lem and characterize the controls under the filtration G such that the additionalinformation for an insider does not blow up the value of the problem.

Let π(t) be the fraction of the wealth invested in the stock (risky asset) at timet by an insider. Since π(·) is a G-adapted process, it is natural to use the forwardintegration to make the integrals well defined. The corresponding wealth processX(c,π)(·) of the insider is given by

d−X(c,π)(t) = X(c,π)(t−)[r(t) + (µ(t)− r(t))π(t)dt+ σ(t)π(t)d−B(t)

+π(t)∫

R0

γ(t, z)N(d−t, dz)]− c(t)dt

with initial value X(c,π)(0) = x.We assume that the agent has a logarithmic utility function. It is convenient to

use such functions because it has iso-elastic marginal utility which means that anagent has the same relative risk-tolerance as toward the end of his life. Moreover,as X(t) ≥ 0 for all t ∈ [0, T ], we can assume that the insider has a relativeconsumption rate λ(·) defined by

λ(t) :=c(t)

X(c,π)(t)

without loss of generality. If X(c,π)(T ) = 0, we put λ(T ) = 0. Then the corre-sponding wealth dynamic of the insider in terms of relative consumption rate λand portfolio π can be rewritten by the following forward S.D.E.

d−X(λ,π)(t) = X(λ,π)(t−)[r(t)− λ(t) + (µ(t)− r(t))π(t)dt+σ(t)π(t)d−B(t) + π(t)

∫

R0

γ(t, z)N(d−t, dz)], (3.1)

for all t ∈ [0, T ], where the initial wealth is X(λ,π)(0) = x > 0.

Problem 3.1. Find the optimal relative consumption rate λ∗(·) and the optimalportfolio π∗(·) for an insider subject to his budget constraint, i.e. find the pair(λ∗, π∗) ∈ A which maximizes the performance function given by

J(λ∗, π∗) := sup(λ,π)∈A

E

[∫ T

0

e−δ(t) ln(λ(t)X(λ,π)(t))dt+Ke−δ(T ) lnX(λ,π)(T )

].

Here, δ(t) ≥ 0 is the discount rate for all t ∈ [0, T ]. It is a given boundeddeterministic function satisfying

∫ T

0e−δ(u)du + Ke−δ(T ) 6= 0. T > 0 is a fixed

terminal time and X(λ,π)(·) is the wealth process satisfying the equation (3.1).


A is the set of all admissible control pairs which will be defined in the followingdefinition.

Definition 3.2. A G-adapted stochastic process pair (λ, π) is called admissible if

(i)∫ T

0

λ(s)ds <∞ a.s..

(ii) π(t), t ∈ [0, T ] is caglad.(iii) π(·)σ(·) and π(·)γ(·, z) are forward integrable with respect to Brownian

motion and the compensated Poisson random measure, respectively.(iv) 1 + π(t)γ(t, z) > επ for a.a. (t, z) with respect to dt × νF (dz), for some

επ ∈ (0, 1) depending on π.

(v)∫ T

0

|(µ(s)− r(s))π(s)|+ σ2(s)π(s)2 +

∫

R0

π(s)2γ(s, z)2νF (dz)ds <∞

almost surely.

(vi) E

[∫ T

0

e−δ(s)| lnλ(s)|ds+Ke−δ(T )| lnX(λ,π)(T )|]<∞.

4. Characterization of the Optimal Consumptionand Investment Choice

Since we use an iso-elastic utility function, this optimal consumption rate de-pends only on the discount rate. The other parameters in the economy such asinterest rates or volatility do not appear. However, the consumption of the agentis depending on these coefficients through the wealth. The next step is to showthat the optimal consumption rate found in Øksendal [25] is also optimal for Prob-lem 3.1, i.e., the relative consumption rate can be chosen independently from theoptimal portfolio if the terminal wealth is added to the problem when the agenthas logarithmic utility function. This result for non-anticipative information wasinitially established by Samuelson [30].

Theorem 4.1. Define λ as

λ(t) :=e−δ(t)

∫ T

te−δ(s)ds+K e−δ(T )

; t ≥ 0. (4.1)

Then λ is an optimal relative consumption rate independent of the portfolio cho-sen, in the sense that (λ, π) ∈ A and

J(λ, π) ≥ J(λ, π)

for all λ and π such that (λ, π) ∈ A.

Proof. The proof can be shown using the additive separability of log-utility func-tion. ¤

Note that since the optimal relative consumption rate, λ∗ = λ does not dependon the portfolio choice, the main problem turns to :


Problem 4.2. Find π∗ such that (λ∗, π∗) ∈ A and

J(λ∗, π∗) := J(π∗)

= sup(λ∗,π)∈A

E

[∫ T

0

e−δ(t) ln(λ∗(t)X(λ∗,π)(t))dt+Ke−δ(T ) lnX(λ∗,π)(T )

].

For all (λ, π) ∈ A, let us define Mπ and Yπ(t) as follows:

Mπ(t) :=∫ t

0

µ(s)− r(s)− σ(s)2π(s)−

∫

R0

π(s)γ(s, z)2

1 + π(s)γ(s, z)νF (dz)

ds

+∫ t

0

σ(s)d−B(s) +∫ t

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

N(d−s, dz) (4.2)

and

Yπ(t) :=∫ t

0

e−δ(s)Mπ(s)ds+Mπ(t)( ∫ T

t

e−δ(s)ds+Ke−δ(T )), (4.3)

for all t ∈ [0, T ]. Moreover, we make the following assumptions:

Assumption 4.3.

(A.1) ∀(λ, π), (λ, β) ∈ A with β bounded, there exists positive τ such that thefamily |Mπ+εβ(T ) |0≤ε≤τ is uniformly integrable.

(A.2) For all t ∈ [0, T ] the process pair (λ, π) where π(s) := χ(t,t+h](s)β0(ω),with h > 0 and β0(ω) being a bounded Gt-measurable random variable,belongs to A.

The following theorem plays a crucial role to obtain the necessary conditionsfor the optimal consumption and investment choice.

Theorem 4.4. Suppose (λ∗, π∗) ∈ A is optimal for Problem 3.1. Then Yπ∗(·)defined in (4.3) is a G-martingale.

Proof. Suppose that (λ∗, π∗) ∈ A is optimal for the insider. We can choose β(·)such that (λ∗, β) ∈ A. Then (λ∗, π∗(·) + yβ(·)) ∈ A, for all y small enough. SinceJ(λ∗, π∗ + yβ) also denoted as J(π∗ + yβ) is maximal at π∗, we have

d

dyJ(π∗ + yβ)|y=0 = 0,

which implies

E

[∫ T

0

e−δ(s)

(∫ s

0

β(u)µ(u)− r(u)− π∗(u)σ(u)2

−∫

R0

(γ(u, z)− γ(u, z)1 + π∗(u)γ(u, z)

)νF (dz)du+∫ s

0

β(u)σ(u)d−B(u)

+∫ s

0

∫

R0

β(u)γ(u, z)1 + π∗(u)γ(u, z)

N(d−u, dz))ds

(4.4)


+Ke−δ(T )

(∫ T

0

β(u)σ(u)d−B(u) +∫ T

0

∫

R0

β(u)γ(u, z)1 + π∗(u)γ(u, z)

N(d−u, dz)

+∫ T

0

β(u)µ(u)− r(u)− π∗(u)σ(u)2

−∫

R0

(γ(u, z)− γ(u, z)1 + π∗(u)γ(u, z)

)νF (dz)du)]

= 0. (4.5)

Let us fix t ∈ [0, T ) and h > 0 such that t+ h ≤ T . We can choose β of the form

β(s) := χ(t,t+h](s)β0 (4.6)

where β0 is a bounded Gt-measurable random variable. Rewriting equation (4.5)by using the equation (4.6), we obtain :

E

[β0

∫ t+h

t

e−δ(s)

(∫ s

t

µ(u)− r(u)− π∗(u)σ(u)2−∫

R0

π∗(u)γ(u, z)2

1 + π∗(u)γ(u, z)νF (dz)du

+∫ s

t

σ(u)d−B(u) +∫ s

t

∫

R0

γ(u, z)1 + π∗(u)γ(u, z)

N(d−u, dz))ds

+ β0

∫ T

t+h

e−δ(s)

(∫ t+h

t

σ(u)d−B(u) +∫ t+h

t

∫

R0

γ(u, z)1 + π∗(u)γ(u, z)

N(d−u, dz)

+∫ t+h

t

µ(u)− r(u)− π∗(u)σ(u)2 −

∫

R0

π∗(u)γ(u, z)2

1 + π∗(u)γ(u, z)νF (dz)

du

)ds

]

+ E

[Ke−δ(T )β0

(∫ t+h

t

µ(u)−r(u)−π∗(u)σ(u)2−

∫

R0

π∗(u)γ(u, z)2

1+π∗(u)γ(u, z)νF (dz)

du

+∫ t+h

t

σ(u)d−B(u) +∫ t+h

t

∫

R0

γ(u, z)1 + γ(u, z)π∗(u)

N(d−u, dz)

)]= 0.

Let us define Mπ(·) as in equation (4.2), then the above equation turns to :

E

[β0

(∫ t+h

t

e−δ(s)Mπ∗(s)ds+Mπ∗(t+ h)

(∫ T

t+h

e−δ(s)ds+Ke−δ(T )

)

−Mπ∗(t)

(∫ T

t


))]= 0.

Let us define

Nπ∗(t) :=∫ t

0

e−δ(s)Mπ∗(s)ds

and

Pπ∗(t) := Mπ∗(t)

(∫ T

t


).

Hence, we have

E[β0

(Nπ∗(t+ h)−Nπ∗(t) + Pπ∗(t+ h)− Pπ∗(t) | Gt

)]= 0.


By using the equation (4.3), we get

E[β0(Yπ∗(t+ h)− Yπ∗(t))] = 0.

Since this holds for all bounded Gt-measurable β0, we have :

E[Yπ∗(t+ h)|Gt] = Yπ∗(t).

Hence Yπ∗ is a G-martingale. ¤

The initial enlargement of filtration technique is a common methodology used instochastic control problems with anticipative information. The main assumptionin this method is that the conditional distribution of the random time is absolutelycontinuous to a measure. It implies that every F-martingale is a semimartingalewith respect to the enlarged filtration. In this paper, we do not assume thisstrict condition and indeed we have this as a result by using Theorem 4.4 andforward integral techniques. In the following theorem, we show that if there existsadmissible optimal portfolio and consumption choices, then F-Brownian motion isa G-semimartingale.

Theorem 4.5. Suppose γ(t, z) 6= 0 and σ(t) 6= 0 for a.a. (t, z, ω). Suppose thatthere exist optimal relative consumption rate and optimal portfolio (λ∗, π∗) ∈ Afor Problem 3.1. Then

(i) B(·) is a (G, P )-semimartingale. Therefore, there exists an adapted finitevariation process α(·) such that the process B(·) defined as

B(t) = B(t)−∫ t

0

α(s)ds; ∀t ∈ [0, T ]

is a G-Brownian motion.(ii) The process

∫ ·

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

N(d−s, dz)

is a G-semimartingale.(iii) The process ∫ ·

0

∫

R0

γ(s, z)N(ds, dz)

is a G-semimartingale.(iv) The optimal portfolio π satisfies the following equation:∫ t

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

(∫ T

s

e−δ(u)du+Ke−δ(T )

)(νG − νF )(ds, dz)

+∫ t

0

(µ(s)− r(s)− σ(s)2π(s) + σ(s)α(s)−

∫

R0

π(s)γ(s, z)2


)

×(∫ T

s


) ds = 0, ∀t ∈ [0, T ].


(v) The optimal relative consumption rate is given by

λ∗(t) =e−δ(t)

∫ T


; ∀t ∈ [0, T ].

Proof. Let (λ, π) ∈ A be an optimal control choice for the Problem 3.1. Byapplying the Fubini’s Theorem to Yπ(t) in the equation (4.3), we can write:

Yπ(t) =∫ t

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

(∫ T

s


)N(d−s, dz)

+∫ t

0

σ(s)

(∫ T

s


)d−B(s)

+∫ t

0

(µ(s)− r(s)− σ(s)2π(s)−

∫

R0

π(s)γ(s, z)2


)

×(∫ T

s


) ds.

Note that since Ft ⊂ Gt for all t ∈ [0, T ], the Poisson random measure N(dt, dz)has a unique compensator with respect to the enlarged filtration G, say νG(dt, dz)for all (t, z) ∈ [0, T ] × R0. Using the orthogonal decomposition into a continuouspart Y c

π (t) and a discontinuous part Y dπ (t), we get as follows

Y cπ (t) =

∫ t

0

σ(s)

(∫ T

s


)d−B(s)

−∫ t

0

σ(s)α(s)

(∫ T

s


)ds

Y dπ (t) =

∫ t

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

(∫ T

s


)N(d−s, dz)

+∫ t

0

∫

R0

θ(s, z)(νF − νG)(ds, dz),

where α(s) and θ(s, ·) are Gs-measurable processes for all s ∈ [0, T ] such that∫ t

0

∫

R0

θ(s, z)(νF − νG)(ds, dz)−∫ t

0

σ(s)α(s)

(∫ T

s


)ds

=∫ t

0

(µ(s)− r(s)− σ(s)2π(s)−

∫

R0

π(s)γ(s, z)2


)

×(∫ T

s


)ds.

(i) For the continuous part Y cπ (t), we use the fact that

∫ t

0

1

σ(s)( ∫ T

0e−δ(u)du+Ke−δ(T )

)dY cπ (s) = B(t)−

∫ t

0

α(s)ds, t ∈ [0, T ]


is a G-martingale. Then we obtain directly that B(·) is a G-semimartingale.(ii) Since Yπ(·) is a G-martingale, we can easily show that Γ(·) defined as

Γ(t) :=∫ t

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

( ∫ T

s

e−δ(u)du+Ke−δ(T ))N(d−s, dz)

+∫ t

0

σ(s)( ∫ T

s

e−δ(u)du+Ke−δ(T ))d−B(s), t ∈ [0, T ]

is a G-semimartingale. Then,∫ ·

0

( ∫ T

s

e−δ(u)du+Ke−δ(T ))−1

dΓ(s)

=∫ ·

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

N(d−s, dz) +∫ ·

0

σ(s)d−B(s)

is also a G-semimartingale. Finally, using (i) we conclude that∫ ·

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

N(d−s, dz)

is a G-semimartingale.(iii) By (ii) and Remark 2.5, we know that

∫ t

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

(νF − νG)(ds, dz), t ∈ [0, T ]

is of finite variation. Using Hypothesis (iv) of Definition 3.2, it followsthat ∫ t

0

∫

R0

γ(s, z)(νF − νG)(ds, dz), t ∈ [0, T ]

is of finite variation. Note that the G-martingale∫ ·

0

∫

R0

γ(s, z)(N − νG)(ds, dz)

can be written as :∫ t

0

∫

R0

γ(s, z)(N − νG)(ds, dz)

=∫ t

0

∫

R0

γ(s, z)N(ds, dz) +∫ t

0

∫

R0

γ(s, z)(νF − νG)(ds, dz), t ∈ [0, T ]

and since∫0

∫R0γ(s, z)(νF − νG)(ds, dz) is of finite variation then

∫ ·

0

∫

R0

γ(s, z)N(ds, dz)

is a G-semimartingale.


(iv) Let us rewrite Yπ(t) as

Yπ(t) =∫ t

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

( ∫ T

s

e−δ(u)du+Ke−δ(T ))(N − νG)(ds, dz)

+∫ t

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

( ∫ T

s

e−δ(u)du+Ke−δ(T ))(νG − νF )(ds, dz)

+∫ t

0

(µ(s)− r(s)− σ(s)2π(s) + σ(s)α(s)−

∫

R0

π(s)γ(s, z)2


)

×( ∫ T

s

e−δ(u)du+Ke−δ(T ))

ds

+∫ t

0

σ(s)(∫ T

s

e−δ(u)du+Ke−δ(T ))dB(s).

Hence by the martingale representation theorem, we conclude that thefinite variation part is zero, i.e.,

∫ t

0

∫

R0

γ(s, z)1 + π(s)γ(s, z)

(∫ T

s

e−δ(u)du+Ke−δ(T ))(νG − νF )(ds, dz)

+∫ t

0

µ(s)− r(s)− σ(s)2π(s) + σ(s)α(s)−

∫

R0

π(s)γ(s, z)2


×( ∫ T

s

e−δ(u)du+Ke−δ(T ))ds = 0.

¤

Finally, as a Corollary, let us present the results for an uninformed agent:

Corollary 4.6. Suppose Ft = Gt, for all t ∈ [0, T ]. Then the optimal portfolioπ(·) solves the following equation:

µ(t)− r(t)− σ(t)2π(t)−∫

R0

π(t)γ(t, z)2

1 + π(t)γ(t, z)νF (dz) = 0, ∀t ∈ [0, T ]

and the optimal relative consumption rate λ(t) is given by

λ(t) =e−δ(t)

∫ T

te−δ(u)du+K e−δ(T )

; ∀t ∈ [0, T ].

Proof. These results can be directly derived from Theorem 4.5. ¤

5. Examples

In this section, we give two examples to illustrate our results. These exampleswere already treated in other papers for the optimal portfolio choices. The aim ofthis section is to show that our approach is coherent and give the same results asthe ones obtained using enlargement of filtration theory.


Example 5.1. The Brownian motion case.

Suppose that γ(t, z) = 0 and σ(t) 6= 0 for almost all (t, z). We denote by π∗i (t)and π∗h(t) the optimal portfolios for the insider and the uninformed (honest) agent,respectively. By Theorem 4.5, the optimal portfolio π∗i (t) satisfies the followingequation

∫ t

0

(∫ T

s


)µ(s)− r(s)− σ(s)2π∗i (s) + σ(s)α(s)

ds = 0,

for all t ∈ [0, T ]. Then, we obtain an explicit solution for π∗i (t):

π∗i (t) =µ(t)− r(t)σ(t)2

+α(t)σ(t)

, ∀t ∈ [0, T ]

and the optimal relative consumption rate for the insider λ∗i (t) given by :

λ∗i (t) =e−δ(t)

∫ T

te−δ(s)ds+Ke−δ(T )

·

For the uninformed agent, by Corollary 4.6, π∗h(t) and λ∗h(t) are given by :

π∗h(t) =µ(t)− r(t)σ(t)2

, λ∗h(t) =e−δ(t)

∫ T


= λ∗i (t)·

By (i) in Theorem 4.5, B(t) is a Gt-semimartingale then

X

“cλ∗

i,π∗i”

i (t) =X

“cλ∗

h,π∗h”

h (t) exp

12

∫ t

0

α(s)2ds+∫ t

0

α(s)dB(s)

(5.1)

where X

“cλ∗

i,π∗i”

i (t) and X

“cλ∗

h,π∗h”

h (t) are the optimal wealth processes for theinsider and the uninformed agent respectively.Hence,

Ji

(cλ∗i , π

∗i

)= Jh

(cλ∗h , π

∗h

)+

12E

[∫ T

0

e−δ(t)

∫ t

0

α(s)2ds dt

]

+12KE

[e−δ(T )

∫ T

0

α(s)2ds

].

(5.2)

Remark 5.2. Note that although the optimal relative consumption rates are thesame, the optimal consumption rates are not the same among the informed anduninformed agent by equation (5.1).

Proposition 5.3. Let Gt = FBt ∨σ(B(T0)), T0 > T for all t ∈ [0, T ]. Assume that

there exist optimal control choices for the Problem 3.1. Let δ(t) and γ(t, z) areequal to zero for all (t, z) ∈ [0, T ]×R0. Then the additional performance functionof an insider is equal to

12

[(T0 − T ) ln(T0 − T ) + T

]+K

2ln

(T0

T0 − T

).


Proof. If we restrict the enlarged filtration to be Gt = FBt ∨ σ(B(T0)), T0 > T ,

then

α(t) =B(T0)−B(t)

T0 − t·

and by the equation (11), the performance function of the informed agent can bewritten in terms of the performance function of the uninformed one as follows :

Ji

(cλ∗i , π

∗i

)= Jh

(cλ∗h , π

∗h

)+

12

∫ T

0

∫ t

0

1T0 − s

ds+K

2

∫ T

0

1T0 − s

ds

= Jh

(cλ∗h , π

∗h

)+

12

[(T0 − T ) ln(T0 − T ) + T

]+K

2ln

(T0

T0 − T

).

¤Example 5.4. The mixed case.

Suppose that γ(t, z) = z and σ(t) 6= 0 for almost all (t, z). We consider theenlarged filtration G′t = Ft ∨ σ(B(T0), η(T0)), T0 > T and take the following as-sumptions:

(1) The informed agent has access to the filtration Gt such that Ft ⊆ Gt ⊆ G′t,t ∈ [0, T ].

(2) The Levy measure νF is given by νF (ds, dz) = ρδ1(dz)ds where δ1(dz) isthe unit point mass at 1.

(3) η(t) is defined as η(t) = Q(t) − ρt with Q being a Poisson process ofintensity ρ.

Using the results of Di Nunno et al. [7] Section 5, we obtain the following optimalportfolio π∗i (t):

π∗i (t) = π∗h(t) +ζ(t)σ(t)

with

π∗h(t) =µ(t)− r(t)σ(t)2

− ρ

σ(t)2,

ζ(t) =1

2σ(t)

[− µ(t) + r(t) + ρ+ σ(t)α(t)− σ(t)2

+√

(µ(t)− r(t)− ρ+ σ(t)α(t) + σ(t)2)2 + 4σ(t)2θ(t)],

α(t) =E[B(T0)−B(s)|Gs]−

T0 − s,

θ(t) =E[Q(T0)−Q(s)|Gs]−

T0 − s,

where the notation E[...]− denotes the left limit in s.Moreover we have the optimal consumption rates λ∗i (t) and λ∗h(t) for the in-

formed and uninformed agents respectively:

λ∗i (t) =e−δ(t)

∫ T

te−δ(s)ds+Ke−δ(T )

= λ∗h(t).


Substituting these equalities into the wealth process equation and by Theorem 4.5,we can express the optimal wealth process of the informed agent in terms of theoptimal wealth process of the uninformed agent :

X

“cλ∗

i,π∗i”

i (t) = X

“cλ∗

h,π∗h”

h (t) exp∫ t

0

(−1

2ζ(s)2 + ζ(s)α(s)

)ds

+∫ t

0

∫

R0

ln(

1 +zζ(s)σ(s)

σ(s)2 + (µ(s)− r(s)− ρ)z

)νG(ds, dz) +

∫ t

0

ζ(s)dB(s)

+∫ t

0

∫

R0

ln(

1 +zζ(s)σ(s)

σ(s)2 + (µ(s)− r(s)− ρ)z

)(N − νG)(ds, dz)

.

Hence,

Ji

(cλ∗i ,π

∗i

)= Jh

(cλ∗h , π

∗h

)+ E

[∫ T

0

e−δ(t)

∫ t

0

(−1

2ζ(s)2 + ζ(s)α(s)

)ds dt

]

+ E

[∫ T

0

e−δ(t)

∫ t

0

∫

R0

ln(

1 +zζ(s)σ(s)

σ(s)2 + (µ(s)− r(s)− ρ)z

)νG(ds, dz) dt

]

+KE

[e−δ(T )

∫ T

0

(−1

2ζ(s)2 + ζ(s)α(s)

)ds

]

+KE

[e−δ(T )

∫ T

0

∫

R0

ln(

1 +zζ(s)σ(s)

σ(s)2 + (µ(s)− r(s)− ρ)z

)νG(ds, dz)

].

Acknowledgment. We are grateful to our supervisors B. Øksendal, N. Privault,and G. Di Nunno for the useful comments and suggestions. We also thank to M.Pontier and G. W. Weber for their fruitful remarks and suggestions. We wish tothank the Advanced Mathematical Methods for Finance (AMaMeF) programme ofthe European Science Foundation (ESF) for the financial supports.

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Delphine David: Laboratoire d’Analyse et Probabilites, Universite d’Evry-Val-d’Essonne, France


URL: http://perso.univ-lr.fr/ddavid/Home.html

Yeliz Yolcu Okur: Centre of Mathematics for Applications (CMA), University ofOslo, Norway


UNIVERSAL MALLIAVIN CALCULUS IN FOCK ANDLEVY-ITO SPACES

DAVID APPLEBAUM

Abstract. We review and extend Lindsay’s work on abstract gradient anddivergence operators in Fock space over a general complex Hilbert space.Precise expressions for the domains are given, the L2-equivalence of norms isproved and an abstract version of the Ito-Skorohod isometry is established.We then outline a new proof of Ito’s chaos expansion of complex Levy-Itospace in terms of multiple Wiener-Levy integrals based on Brownian mo-tion and a compensated Poisson random measure. The duality transformnow identifies Levy-Ito space as a Fock space. We can then easily obtainkey properties of the gradient and divergence of a general Levy process. Inparticular we establish maximal domains of these operators and obtain theIto-Skorohod isometry on its maximal domain.

1. Introduction

Malliavin calculus is one of the deepest and most important areas within con-temporary stochastic analysis. It was originally developed as a new probabilistictechnique to find smooth densities for solutions of stochastic differential equations(SDEs). At a more fundamental level it provides an intrinsic differential calculusin Gaussian probability spaces based on two mutually adjoint linear operators -the gradient and the divergence (see e.g. [39], [42], [30], [26], [53] for monographaccounts). More recently it has enabled the developments of new techniques inmathematical finance (see [40] and references therein).

Ever since the early days of the subject there has been plenty of activity inwidening the scope of Malliavin calculus to include jump processes and [10] isa monograph dedicated to this theme. More recently there has been increasedinterest in these ideas - partly due to new progress in finding smooth densities forspecial classes of SDEs driven by Levy processes (see e.g. [31], [27]) but also forthe need to extend the calculus to financial models based on jump processes (seee.g. [34], [13], [47] and the forthcoming monograph [17]).

Fock space has long been known to be intimately connected with probabilitytheory. Indeed if the natural L2-space of a process has a chaotic decompositionthen it is automatically isomorphic to a Fock space over the Cameron-Martin space(in probabilistic language) or one-particle space (in physical terms). The first

2000 Mathematics Subject Classification. Primary 60H07; Secondary 81S25, 28C20, 60G51.Key words and phrases. Fock space, exponential vector, universal annihilation and cre-

ation operators, number operator, Lindsay-Malliavin transform, Levy process, multiple Wiener-Levy integrals, Ito representation theorem, chaos decomposition, duality transform, stochastic(Doleans-Dade) exponential, gradient, divergence, Malliavin derivative, Ito-Skorohod isometry.

119



120 DAVID APPLEBAUM

direct use of Fock space ideas in (classical) Malliavin calculus seems to have beenby Dermoune, Kree and Wu [15] in work on non-anticipating stochastic calculus(including a generalised Ito formula) for the Poisson process. These ideas werethen taken up by Nualart and Vives who defined the gradient and divergence forthe Poisson process directly in Fock space [43] and Dermoune [14] who extendedthe work of [15] to general Levy processes.

In a separate development, Hudson and Parthasarathy [25] realised that Fockspace is the natural setting for a quantum stochastic calculus based on a non-commutative splitting of Brownian motion and the Poisson process into constituentannihilation, creation and conservation noise processes (see also [46], [41], [37]).Parthasarathy (see [46] p.155-8) also showed that Levy processes may be repre-sented in Fock space and the corresponding extended quantum stochastic calculuswas developed in [1], [3]. Key papers by Belavkin [6, 5, 7] and Lindsay [36] broughtMalliavin calculus directly into the non-commutative framework to devise a non-anticipating quantum stochastic calculus. More recent developments in this areacan be found in [4]. A fully quantised Malliavin calculus based on the Wignerdensity is due to Franz, Leandre and Schott [21], [22].

The goal in the first part of this paper is to begin to develop a universal Malli-avin calculus in Fock space over a general separable Hilbert space (see also Privaultand Wu [51]). There is no probability content (either classical or quantum) in thetheory at this stage. We work in an abstract Hilbert space and we focus ourstudies on two operators originally introduced by Lindsay [36] in the context ofa Fock-Guichardet space [24] and called abstract gradient and divergence therein.We prefer to call them universal annihilation and creation operators as they can betransformed into the usual annihilation and creation operators indexed by a givenvector in one-particle space after composition with a suitable Dirac bra or ket op-erator. Following Privault [50] (see also Privault and Wu [51]) we denote these by∇− and ∇+ respectively. The aim of universal Malliavin calculus can be summedup succinctly as follows - given a process having a chaos decomposition, map ∇−and ∇+ unitarily into the L2-space of the process where they become the gradientD and divergence δ. Then structural properties of ∇− and ∇+ are automaticallytransferred to D and δ with little additional effort. In this paper we illustrate thistechnique through application to a Levy process, but it could easily be applied toany other process having a chaotic representation, such as the Azema martingale[20] or the Dunkl process [23].

The results obtained for ∇− and ∇+ in [36] were extended to Fock space overan abstract separable Hilbert space by Lindsay in Proposition 3.1 of [37]. Heestablished three key properties of these operators:

(1) The maximal domain of ∇−.(2) The factorisation of the number operator N = ∇+∇−.(3) An isometry-type property for ∇+ which generalises the key Ito-Skorohod

isometry that lies at the heart of non-anticipating stochastic calculus (aresult of this kind was also established independently by Privault and Wu[51]).

The proofs of these results were outlined in [36]. We give full proofs in section3 of this paper for the sake of completeness. The approach presented here is

UNIVERSAL MALLIAVIN CALCULUS 121

different and we reformulate the result of (1) in a way that will be more familiar toprobabilists. We also extend the theory by obtaining a result on the L2-equivalenceof norms which is the first step towards a theory of infinite dimensional Sobolevspaces at this level.

In section 4 we turn to probability theory and study the chaotic representationof a Levy process. This result, originally due to Ito [28], shows that the naturalL2-space H of the process (called Levy-Ito space herein) is naturally isomorphicto the infinite direct sum of the chaoses generated by multiple Wiener-Levy inte-grals constructed from the Brownian motion B and compensated Poisson measureN associated to the process through its Levy-Ito decomposition. More straight-forward proofs of this result have recently been found by Løkka [38] for squareintegrable processes and Petrou [47] in the general case. Their approach is to iter-ate the Ito representation of the process whereby any element of H is a constantplus an Ito stochastic integral with respect to B and N . The Ito representationis itself proved by a density argument using a class of exponential martingales ofWiener-Levy stochastic integrals of deterministic functions. We briefly outline ageneralisation of these results. The main difference for us is that H is complexand this allows a simpler proof of the Ito representation using a more natural classof exponential martingales. We only give outline proofs here as the methodologyis well-known and a full account will appear shortly in [2] (ii) (similar ideas areemployed in Bichteler [9], p.259.)

The chaotic representation induces a unitary isomorphism called the dualitytransform between H and a certain Fock space. In section 4 we also prove thatthe image of the exponential vectors under this isomorphism is the stochastic orDoleans-Dade exponentials. A general result of this type was hinted at by Meyerin [41] p.71 but he only explicitly considered the Wiener space case. These vectorsand their chaos expansions have recently found interesting applications to interest-rate modelling [11].

In the last part of the paper we apply the duality transform to the results (1)to (5) of section 1 to obtain the Levy process versions of these for D and δ. Ourmain results are

(1) The maximal domains of D and δ expressed in terms of chaos expansions.(2) The full Ito-Skorohod isometry on a maximal domain.

We note that the Ito-Skorohod isometry has also recently been investigatedin [16] using white noise analysis techniques but these authors were restricted tousing a pure jump square integrable Levy process without drift and no explicitdomain was given. A white noise approach is also developed in [33] but under theconstraint that the Levy measure has moments to all orders (see also [32]).

In this paper we have only made the first few steps in the direction of a universalMalliavin calculus. A defect of the theory as it stands is that it only works atthe Hilbert space level and so there are, for example, no direct analogues of thefull range of infinite dimensional Sobolev spaces which require Lp structure whenp 6= 2. It may be that this can be remedied by using a Banach-Fock space as insection 6 of [8].

Some related work on Malliavin calculus for Levy processes has recently beenpresented in [54]. The key novel ingredient here is the development of a new

122 DAVID APPLEBAUM

canonical construction for pure jump Levy processes which facilitates the study ofthe Malliavin derivative through its representation as a difference quotient.

Notation: All inner products in complex Hilbert spaces are conjugate linearon the left. The algebraic tensor product of two vector spaces V1 and V2 will bedenoted V1⊗V2. If T is a closable operator in a Hilbert space, we will throughoutthis paper use the same notation T for its closure on the larger domain. Dom(T )will always denote the maximal domain of T . If S is a topological space then B(S)is its Borel σ-algebra.

2. Fock Space Preliminaries

In this section we define some key operators in Fock space. Full proofs of allresults mentioned here can be found in [46], [41] or [37].

Let H be a complex separable Hilbert space and H⊗n

be its n-fold tensorproduct. We denote by H⊗n

s the closed subspace of H⊗n

comprising symmetrictensors. H¯n

s is the image of H⊗n

s under the bijection ψ →√n!ψ and is regarded

as a Hilbert space with respect to the inner product 〈·, ·〉H¯ns

= n!〈·, ·〉H⊗ns. We

write f¯n

:=√n!f⊗

n

. If g ∈ H we define the symmetrisation of f¯n

and g to bethe vector Symm(f¯

n

, g) ∈ H¯n+1

s defined by

Symm(f¯n

, g) :=

√n!

n+ 1

n∑r=0

f⊗n−r ⊗ g ⊗ f⊗

r

.

Note that the choice of normalisation ensures that Symm(f¯n

, f) = f¯n+1

.Symmetric Fock space over H is Γ(H) :=

⊕∞n=0H

⊗n

s , where by conventionH⊗0

s := C and H⊗1

s := H. We also define Γ(H) :=⊕∞

n=0H¯n

s . Of course Γ(H)and Γ(H) are naturally isomorphic (see below) and we will freely move betweenthese spaces in the sequel. We will often identify H⊗n

s with its natural embeddingin Γ(H) whereby each fn ∈ H⊗n

s is mapped to (0, . . . , 0, fn, 0, . . .).The exponential vector e(f) ∈ Γ(H) corresponding to f ∈ H is defined by

e(f) =(1, f, f⊗f√

2!, . . . , f⊗

n

√n!, . . .

). We have 〈e(f), e(g)〉 = e〈f,g〉, for each f, g ∈ H.

The mapping f → e(f) from H to Γ(H) is analytic.If D ⊆ H we define E(D) to be the linear span of e(f), f ∈ D. In particular

if D is a dense linear manifold in H then E(D) is dense in Γ(H). We defineE := E(H). If f ∈ H, the corresponding annihilation operator a(f), creationoperator a†(f), exponential annihilation operator U(f) and exponential creationoperator U†(f) are defined on E by linear extension of the following prescriptions:

a(f)e(g) = 〈f, g〉e(g), (2.1)

a†(f)e(g) =d

dte(g + tf)

∣∣∣∣t=0

, (2.2)

U(f)e(g) = e〈f,g〉e(g), (2.3)

U†(f)e(g) = e(g + f), (2.4)


for all g ∈ H. Each of these operators is closable, indeed a†(f) ⊆ a(f)∗, U†(f) ⊆U(f)∗, for each f ∈ H. We also have the canonical commutation relations

a(f)a†(g)ψ − a†(g)a(f)ψ = 〈f, g〉ψ, (2.5)

for all f, g ∈ H,ψ ∈ E .If T is a contraction in H then its second quantisation Γ(T ) is the contraction

in Γ(H) whose action on E is given by linear extension of

Γ(T )e(f) = e(Tf). (2.6)

In particular if T is unitary, then so is Γ(T ). If A is a bounded self-adjointoperator in H we define the associated conservation operator Λ(A) to be theinfinitesimal generator of the one-parameter unitary group (Γ(eitA), t ∈ R). It iseasily checked that E ⊆ Dom(Λ(A)) and we have the useful identity

〈e(f),Λ(A)e(g)〉 = 〈f,Ag〉〈e(f), e(g)〉, (2.7)

for all f, g ∈ H.The number operator is defined by N := Λ(I). Its domain is Dom(N) =

(fn, n ∈ Z+) ∈ Γ(H);∑∞

n=1 n2||fn||2 <∞

. The associated contraction semi-group is (Tt, t ≥ 0) where for each t ≥ 0

Tt := e−tN =∞∑

n=0

e−tnPn,

and where Pn denotes the orthogonal projection from Γ(H) to H⊗n

s .The unitary isomorphism from Γ(H) to Γ(H) which we employ to identify these

spaces is (N !)−12 .

Using the analyticity of exponential vectors, we can easily check that finiteparticle vectors are in the domains of annihilation and creation operators and wededuce the following:

a(f)g¯n

= n〈f, g〉g¯n−1, (2.8)

a†(f)g¯n

= Symm(g¯n

, f), (2.9)

for all f, g ∈ H.If H = H1 ⊕ H2 we may identify Γ(H) with Γ(H1) ⊗ Γ(H2) via the natural

isomorphism which maps e(f) to e(f1) ⊗ e(f2) for each f = (f1, f2) ∈ H. In thiscontext we will always denote the linear span of the exponential vectors in Hi byEi(i = 1, 2).

3. Universal Annihilation and Creation Operators

For each t ∈ R we define linear operators Vt : Γ(H) → Γ(H) ⊗ Γ(H) andV †t : Γ(H) ⊗ Γ(H) → Γ(H) on the dense domains E and E ⊗ E (respectively) bylinear extension of the following prescriptions:

Vte(f) = e(f)⊗ e(tf), (3.1)

124 DAVID APPLEBAUM

V †t (e(f)⊗ e(g)) = U(tg)†e(f) = e(f + tg), (3.2)for all f, g ∈ H.

Proposition 3.1. For each t ∈ R, Vt and V †t are closable with V †t ⊆ V ∗t .

Proof. The result will follow if we can show that these operators are mutuallyadjoint. Now for all f, g, h ∈ H, t ∈ R,

〈Vte(f), e(g)⊗ e(h)〉 = 〈e(f)⊗ e(tf), e(g)⊗ e(h)〉= 〈e(f), e(g)〉〈e(tf), e(h)〉= 〈et〈h,f〉e(f), e(g)〉= 〈U(th)e(f), e(g)〉 (by 2.3)

= 〈e(f), U(th)†e(g)〉= 〈e(f), V †t e(g)⊗ e(h)〉.

¤

We define linear operators ∇− : Γ(H) → Γ(H)⊗H and ∇+ : Γ(H)⊗H → Γ(H)on the dense domains E and E⊗H (respectively) by linear extension of the followingprescriptions, for each f, g ∈ H:

∇−e(f) =d

dtVte(f)

∣∣∣∣t=0

,

so by (3.1)∇−e(f) = e(f)⊗ f, (3.3)

and ∇+(e(f)⊗ g) =d

dtV †t e(f)⊗ e(g)

∣∣∣∣t=0

,

so by (3.2)∇+(e(f)⊗ g) = a†(g)e(f). (3.4)

Proposition 3.2. ∇− and ∇+ are closable with ∇+ ⊆ (∇−)∗.

Proof. The fact that ∇− and ∇+ are mutually adjoint follows from differentiationof the adjunction relation between Vt and V †t and the result follows. ¤

We call ∇− and ∇+ universal annihilation and universal creation operators(respectively) for as will be shown below (Proposition 3.5) they generate all of the“usual” creation and annihilation operators which depend on a choice of vectorin H. For each n ∈ Z+ we denote the restrictions of ∇− and ∇+ to H¯n

s andH¯n

s ⊗ H (respectively) by ∇−n and ∇+n . Using the analyticity of exponential

vectors, we can easily deduce that Ran(∇−n ) ⊆ H¯n

s ⊗H and Ran(∇+n ) ⊆ H¯n+1

s

and obtain the following analogues of (2.8) and (2.9) (c.f. [50]):

∇−n f¯n

= nf¯n−1 ⊗ f (3.5)

∇+n f

¯n ⊗ g = Symm(f¯n

, g), (3.6)


for each f, g ∈ H.

Lemma 3.3. For each n ∈ Z+, ∇−n and ∇+n are bounded operators with ||∇−n || =√

n and ||∇+n || =

√n+ 1.

Proof. For each f ∈ H,

||∇−n f¯n || = n||f¯n−1 ⊗ f ||

= n√

(n− 1)!||f ||n =√n||f¯n ||H¯n

s.

The result for ∇−n follows from the fact that the linear span of f¯n

, f ∈ H isdense in H¯n

s .For each f, g ∈ H,

||∇+n (f¯

n ⊗ g)|| =

∣∣∣∣∣

∣∣∣∣∣

√n!

n+ 1

n∑r=0

f⊗n−r ⊗ g ⊗ f⊗

r

∣∣∣∣∣

∣∣∣∣∣≤

√(n+ 1)!||f ||n||g||

=√n+ 1||f¯n ⊗ g||H¯n

s ⊗H .

Since the linear span of f¯n ⊗ g, f, g ∈ H is dense in H¯n

s ⊗H, it follows that∇+

n is bounded. To see that the bound is obtained, observe that for all f ∈ H,

∇+n (f¯

n ⊗ f) = Symm(f¯n

, f) =√n+ 1f¯

n ⊗ f.

¤

If φn ∈ H¯n

s ⊗H, there exist sequences (fn,r, r ∈ N) and (gr, r ∈ N) where eachfn,r ∈ H¯n

s and gr ∈ H such that φn =∑∞

r=1 fn,r ⊗ gr. We define

φn := ∇+nφn =

∞∑r=1

Symm(fn,r, gr).

In the next theorem we will find it convenient to identify Γ(H)⊗Hwith

⊕∞n=0(H

¯n

s ⊗H).

Theorem 3.4. (1)

Dom(∇−) =ψ = (ψn, n ∈ Z+) ∈ Γ(H);

∞∑n=1

nn!||ψn||2 <∞.

(2) Dom(∇+) =φ = (φn, n ∈ Z+) ∈ Γ(H)⊗H;

∑∞n=0 ||φn||2 <∞

.

Proof. (1) (Sufficiency)

Let ψ ∈ Γ(H) be such that∑∞

n=1 nn!||ψn||2 <∞ and for each M ∈ Z+,define ψ(M) = (ψ0, ψ1, . . . , ψM , 0, 0, . . .). Clearly ψ(M) → ψ as M → ∞.Using Lemma 3.3 we see that for each M,N ∈ N, N > M

||∇−ψN −∇−ψM ||2 =N∑

n=M+1

nn!||ψn||2 → 0 as N,M →∞.

126 DAVID APPLEBAUM

Hence (∇−ψN , N ∈ N) converges to a vector φ in Γ(H)⊗H. Since ∇− isclosed, we deduce that φ = ∇−ψ and so ψ ∈ Dom(∇−).

(Necessity) Suppose that ψ = (ψn, n ∈ Z+) ∈ Dom(∇−), then ∇−ψ =(∇−nψn, n ∈ Z+) and again using Lemma 3.3 we obtain

||∇−ψ||2 =∞∑

n=0

||∇−nψn||2

=∞∑

n=1

nn!||ψn||2 <∞.

(2) This is proved by exactly the same argument as (1).¤

Clearly, as an operator on Γ(H), Dom(∇−) = Dom(√N) (see [37], [36]). To

some extent, Theorem 3.4 (2) tells us less than (1) (although it is sufficient forapplications in probability). Another approach to ∇+ is given by J.M.Lindsayin [37]. Let Φ(H) be the full Fock space over H. Then since Γ(H) ⊗ H =⊕∞

n=0(H⊗n

s ⊗H) ⊆ Φ(H) we can regard ∇+ as an operator from Φ(H) to Γ(H).In fact it is not difficult to verify that in this case ∇+ =

√NPs where Ps is the

orthogonal projection from Φ(H) to Γ(H).

Now suppose that H = H1 ⊕H2, then as previously remarked we may identifyΓ(H) with Γ(H1)⊗Γ(H2). We may also identify Γ(H)⊗H with [(Γ(H1)⊗H1)⊗Γ(H2)] ⊕ [Γ(H1) ⊗ (Γ(H2) ⊗ H2)] in an obvious way. For i = 1, 2 ∇±i denotesthe universal annihilation/creation operators associated to each Hi and πi arethe isometric embeddings of Hi into H1 ⊕ H2, so for example if f ∈ H1 thenπ(f) = (f, 0). Note that π∗1((f1, f2)) = f1 for all fi ∈ Hi. For simplicity, we willcontinue to use the notation πi and π∗i when these operators are tensored with theidentity to act in tensor products.

Using the identifications given above, it is not difficult to verify that

∇− = π1(∇−1 ⊗ I) + π2(I ⊗∇−2 ), (3.7)

on Dom(∇−1 )⊗Dom(∇−2 ), and

∇+ = (∇+1 ⊗ I)π∗1 + (I ⊗∇+

2 )π∗2 , (3.8)

on Dom(∇+1 )⊕Dom(∇+

2 ) (c.f. [46], p.150).

It is useful to think of ∇− as a “gradient” and ∇+ as a “divergence” and wewill make these correspondences precise later. In this respect, we should defineassociated “directional derivatives”. For this purpose we introduce the Dirac “bra”and “ket” operators εf : H → C and ε†f : C→ H by

εf (g) = 〈f, g〉, ε†f (α) = αf,

for each f, g ∈ H,α ∈ C (c.f. [18]). These operators are clearly linear, boundedand mutually adjoint with each ||εf || = ||ε†f || = ||f ||. The nature of the “directionalderivative” operators is revealed in the following result:


Proposition 3.5. For each f ∈ H,(1) (I ⊗ εf ) ∇− = a(f),(2) ∇+ (I ⊗ ε†f ) = a†(f), 1

on E(H).

Proof. (1) For each f, g ∈ H,

(I ⊗ εf ) ∇−e(g) = (I ⊗ εf )(e(g)⊗ g) = 〈f, g〉e(g) = a(f)e(g).

(2) follows by taking adjoints in (1).¤

Using a density argument, it is easily verified that Proposition 3.5 (1) extendsto Dom(∇−). Moreover it follows that Dom(∇−) is the maximal domain for alla(f), f ∈ H.

We will now give a noncommutative factorisation of the number operator (see[50]) for a different factorisation in the additive sense). First, for each n ∈ N, wedefine a linear operator Wn from H¯n

to H¯n−1 ⊗H by

Wn =1√n∇−n .

We also define W : Γ(H) → Γ(H)⊗H by W =⊕∞

n=0Wn where W0 := 1.

Theorem 3.6. (1) Wn is unitary for each n ∈ Z+.(2) W is unitary.(3) On their maximal domains,

∇− = W√N = (

√N + 1⊗ I)W,

∇+ =√NW ∗ = W ∗(

√N + 1⊗ I).

(4) On Dom(N),N = ∇+∇−.

Proof. (1) Let f ∈ H then Wnf¯n

=√nf¯

n−1 ⊗ f and so

||Wnf¯n || =

√n||f¯n−1 ||.||f ||

=√n!||f ||n = ||f¯n ||.

Hence Wn is an isometry between total sets and so extends to a unitaryoperator by linearity and continuity.

(2) Follows immediately from (1).(3) ∇− = W

√N on Dom(∇−) = Dom(

√N) is immediate. SinceW ∗

n = 1√n∇+

n

it follows that W ∗ = N− 12∇+ and hence W ∗(Dom(∇+)) ⊆ Dom(

√N). So

∇+ =√NW ∗ on Dom(∇+). The other results are proved similarly.

(4) This follows immediately from (3) and (2).¤

1Here we are identifying H with H ⊗ C.

128 DAVID APPLEBAUM

Dom(∇−) becomes a complex Hilbert space with respect to the inner product〈·, ·〉1 where for each ψ1, ψ2 ∈ Dom(∇−),

〈ψ1, ψ2〉1 = 〈ψ1, ψ2〉+ 〈∇−ψ1,∇−ψ2〉.

Consider the self-adjoint linear operator Q := (1+N)−12 = π−

12

∫∞0t−

12 e−tTtdt,

(see Lemma 3.12 in [26] for the last identity).It is easy to check thatQ is a bounded operator on Γ(H) and thatQDom(∇−) ⊆

Dom(N).

Theorem 3.7. Q is a unitary isomorphism between (Γ(H), ||·||) and (Dom(∇−), ||·||1).Proof. (c.f. the proof of Proposition 3.14 in [26]). Let ψ1, ψ2 ∈ Dom(∇−), thenby Theorem 3.6

〈ψ1, ψ2〉 = 〈Q−1Qψ1, Q−1Qψ2〉

= 〈Q−2Qψ1, Qψ2〉= 〈Qψ1, Qψ2〉+ 〈NQψ1, Qψ2〉= 〈Qψ1, Qψ2〉+ 〈∇−Qψ1,∇−Qψ2〉= 〈Qψ1, Qψ2〉1.

Hence Q is an isometric embedding of (Γ(H), || · ||) in (Dom(∇−), || · ||1). Theresult follows from the fact that Ran(Q) is dense in (Dom(∇−), || · ||1). To see thislet D be the linear space comprising those sequences (ψn, n ∈ N) ∈ Γ(H) whereψn = 0 for all but finitely many n. Clearly D is dense in Dom(∇−). HoweverQD = D and the result follows.

¤

From now on we will use Ξ to denote the Hilbert space Dom(∇−) equippedwith the inner product 〈·, ·〉1. In the sequel we will also want to work with theclosed linear operator ∇−⊗I acting in Γ(H)⊗H. The Hilbert space Dom(∇−⊗I)equipped with the graph norm is precisely Ξ⊗H.

Let τ be the tensor shift on H⊗H so that τ is the closed linear extension of themap τ(f ⊗ g) = g ⊗ f for each f, g ∈ H. It is easily verified that τ is self-adjointand unitary (in physicists’ language, τ is an example of a “parity operator”).

Our next result is an abstract version of the “Ito-Skorohod isometry”. Notehowever that the presence of the operator τ ensures that the isometry propertydoes not in fact hold.

Theorem 3.8. For all ψi ∈ Ξ⊗H(i = 1, 2)

〈∇+ψ1,∇+ψ2〉 = 〈ψ1, ψ2〉+ 〈(I ⊗ τ)(∇− ⊗ I)ψ1, (∇− ⊗ I)ψ2〉. (3.9)

Furthermore ∇+ is a contraction from Ξ⊗H into Γ(H).

Proof. To establish (3.9), let fi, gi ∈ H(i = 1, 2). We then find that for eachn ∈ N,


〈∇+n f

¯n

1 ⊗ g1,∇+n f

¯n

2 ⊗ g2〉

=n!

n+ 1

n∑r=0

n∑s=0

〈f⊗n−r

1 ⊗ g1 ⊗ f⊗r

1 , f⊗n−s

2 ⊗ g2 ⊗ f⊗s

2 〉

= 〈f¯n

1 ⊗ g1, f¯n

2 ⊗ g2〉+ n2〈f¯n−1

1 , f¯n−1

2 〉〈g1, f2〉〈f1, g2〉= 〈f¯n

1 ⊗ g1, f¯n

2 ⊗ g2〉+ 〈(I ⊗ τ)(∇−n ⊗ I)f¯

n

1 ⊗ g1, (∇−n ⊗ I)f¯n

2 ⊗ g2〉.The required result follows from here by linearity and continuity. To establish thecontraction property, we have from (3.9) that for all ψ ∈ Ξ⊗H,

||∇+ψ||2 = ||ψ||2 + 〈(I ⊗ τ)(∇− ⊗ I)ψ, (∇− ⊗ I)ψ〉,and the result follows easily from this by using the Cauchy-Schwarz inequality andthe isometry property of I ⊗ τ. ¤

The last result is closely related to the canonical commutation relations (2.5).This is not so clear from the argument in the proof of Theorem 3.8 however it isinstructive to compute actions on exponential vectors. If fi, gi ∈ H(i = 1, 2), wefind that

〈∇+e(f1)⊗ g1,∇+e(f2)⊗ g2〉= 〈a†(g1)e(f1), a†(g2)e(f2)〉= 〈e(f1), e(f2)〉〈g1, g2〉+ 〈a(g2)e(f1), a(g1)e(f2)〉= 〈e(f1)⊗ g1, e(f2)⊗ g2〉+ 〈(I ⊗ τ)(∇− ⊗ I)e(f1)⊗ g1, (∇− ⊗ I)e(f2)⊗ g2〉.

Now let H = L2(S,Σ, µ) where S is a locally compact topological space, Σis its Borel σ-algebra and µ is a Borel measure defined on (S,Σ). In this caseΓ(H)⊗H = L2(S,Σ, µ; Γ(H)). If ψn ∈ H¯n

, ψn−1(·, s) will denote the symmetricfunction of n − 1 variables obtained by fixing s ∈ S. We then have for all ψ =(ψn, n ∈ Z+) ∈ Dom(∇−),

||∇−ψ||2 =∫

S

||∇−s ψ||2Γ(H)µ(ds),

where for µ-almost all s ∈ S∇−s ψ := (nψn−1(·, s), n ∈ N). (3.10)

We call ∇−s ψ the Lindsay-Malliavin transform of ψ at s. Note that ∇−s is nota bona fide operator since the right hand side of (3.10) depends on the choice of asequence of functions from the equivalence class of ψ.

If µ is a regular measure (so that compact sets have finite mass) then the spaceD of continuous functions with compact support is a dense subspace of H. In this

130 DAVID APPLEBAUM

case the Lindsay-Malliavin transform is a genuine linear operator whose action onE(D) is given by

∇−s e(f) = f(s)e(f),

for all f ∈ D. So each ∇−s is densely defined on E(D); however it is well knownthat these operators are not closable (see e.g. [35]).

Even when µ fails to be regular we can rewrite the result of Theorem 3.8 by usingthe Lindsay-Malliavin transform. First we observe that Ξ ⊗ H = L2(S,Σ, µ; Ξ)and elements of this space may be regarded as equivalence classes of mappingsfrom S to Ξ. Note that ∇− ⊗ I : L2(S,Σ, µ; Ξ) → L2(S2,Σ⊗

2, µ× µ; Γ(H)).

Corollary 3.9. If X,Y ∈ L2(S,Σ, µ; Ξ), then

〈∇+X,∇+Y 〉 =∫

S

〈X(s), Y (s)〉µ(ds) +∫

S

∫

S

〈∇−t X(s),∇−s Y (t)〉µ(ds)µ(dt).

(3.11)

Proof. We use the same notation as in the proof of Theorem3.8. It is sufficientto consider the case where X(t) = f¯

n

1 g1(t) and Y (t) = f¯n

2 g2(t) for each t ∈ S.Using (3.10) we obtain

〈(I ⊗ τ)(∇−n ⊗ I)f¯n

1 ⊗ g1, (∇−n ⊗ I)f¯n

2 ⊗ g2〉= n2〈f¯n−1

1 , f¯n−1

2 〉〈g1, f2〉〈f1, g2〉= n2〈f¯n−1

1 , f¯n−1

2 〉∫

S

g1(s)f2(s)µ(ds)∫

S

f1(t)g2(t)µ(ds)

=∫

S

∫

S

〈∇−t X(s),∇−s Y (t)〉µ(ds)µ(dt),

and the result follows. ¤

Remark. Some of the main results of this section - Theorems 3.4, 3.6 (4), 3.8and Corollary 3.9 are all given, at least in outline in [37] Propositions 3.1 and 3.2(see also [36] for the Guichardet space version). A similar result to Theorem 3.8is also established in [51] - see Proposition 1 therein.

4. The Chaos Decomposition of Levy-Ito Space

4.1. Preliminaries on Levy Processes [2]. Let (Ω,F , (Ft, t ≥ 0), P ) be astochastic base wherein the filtration (Ft, t ≥ 0) satisfies the usual hypothesesof completeness and right continuity. Let X = (X(t), t ≥ 0) be an adapted real-valued Levy process defined on (Ω,F , P ) so that X(0) = 0 (a.s.), X has stationaryincrements and strongly independent increments (in the sense that X(t)−X(s) isindependent of Fs for all 0 ≤ s < t < ∞), X is stochastically continuous and itspaths are a.s. cadlag. We have the Levy-Khintchine formula

E(eiuX(t)) = e−tη(u),


for all t ≥ 0, u ∈ R, where η : R → C is a continuous, hermitian negative definitemapping for which η(0) = 0. It has the canonical form

η(u) = −ibu+12σ2u2

+∫

R−0(1− eiuy + iuy1B1(y))ν(dy),

where b ∈ R, σ ≥ 0 and ν is a Levy measure on R− 0, i.e. ν is a Borel measurefor which

∫R−0(1 ∧ |y|2)ν(dy) <∞. Information about the sample paths of X is

given by the Levy-Ito decomposition:

X(t) = bt+ σB(t) +∫

|x|<1

xN(t, dx) +∫

|x|≥1

xN(t, dx). (4.1)

Here N is the Poisson random measure on R+ × (R− 0) defined by

N(t, A) := #0 ≤ s ≤ t,∆X(s) ∈ A,for each t ≥ 0, A ∈ B(R−0), N is the compensated random measure defined by

N(t, A) = N(t, A)− tν(A),

and B = (B(t), t ≥ 0) is a standard Brownian motion which is independent of N .

4.2. The Ito Representation Theorem. In this and the next section, webriefly outline proofs of results which are given more fully in [2](ii).

We fix T > 0 and let f ∈ L2([0, T ),R). We may then form the Wiener-Itointegral Xf (t) =

∫ t

0f(s)dX(s), for each 0 ≤ t ≤ T . We define

Mf (t) := expiXf (t) +

∫ t

0

η(f(s))ds.

Lemma 4.1. For each f ∈ L2([0, T ]), u ∈ R, t ∈ [0, T ],

(1) E(eiuXf (t)) = exp− ∫ t

0η(uf(s))ds

.

(2) (Mf (t), t ∈ [0, T ]) is a complex-valued square-integrable martingale withstochastic differential

dMf (t) = iσf(t)Mf (t−)dB(t) + (eif(t)x − 1)Mf (t−)N(dt, dx). (4.2)

Proof. These are both straightforward applications of Ito’s formula applied to theprocesses (eiuXf (t), 0 ≤ t ≤ T ) and (Mf (t), 0 ≤ t ≤ T ), respectively. ¤

From now on, for each 0 ≤ t ≤ T , we require that Ft = σX(s), 0 ≤ s ≤t. We define Levy-Ito space to be the complex separable Hilbert space H :=L2(Ω,FT , P ;C). PT will denote the predictable σ-algebra generated by processesdefined on [0, T ]× Ω.

Lemma 4.2. Mf (T ), f ∈ L2([0, T ]) is total in H.

Proof. This is a consequence of the injectivity of the Fourier transform. The proofis similar to that of Lemma 4.3.2 in [45]. ¤

132 DAVID APPLEBAUM

Let H(B)2 (T ) be the complex Hilbert space of all complex predictable processes

satisfying∫ T

0E(|F (t)|2)dt < ∞ and H(N)

2 (T ) be the complex Hilbert space of allPT × B(R− 0) measurable mappings G : [0, T ]× (R− 0)× Ω → C for which∫ T

0

∫R−0 E(|G(t, x)|2)ν(dx)dt <∞.

Theorem 4.3. [The Ito Representation]If F ∈ H, then there exists unique ψ0 ∈ H(B)

2 (T ) and ψ1 ∈ H(N)2 (T ) such that

F = E(F ) + σ

∫ T

0

ψ0(s)dB(s) +∫ T

0

∫

R−0ψ1(s, x)N(ds, dx). (4.3)

Proof. The result holds for F = Mf (T ) by (4.2) and is easily extended to finitelinear combinations of such random variables. The extension to arbitrary F isby approximation using Lemma 4.2 (see the proof of Theorem 4.3.3. in [45],Proposition 3 in [38]) and Proposition 2.1 in [47].) ¤4.3. Multiple Wiener-Levy Integrals and the Chaos Decomposition. LetX be a Levy process with associated Levy-Ito decomposition (4.1). Let S =[0, T ] × R. We consider the associated martingale-valued measure M defined on(S, I) by the prescription

M([0, t]×A) = N(t, A− 0) + σB(t)δ0(A)for each t ∈ [0, T ], A ∈ B(R) where I is the ring comprising finite unions of sets

of the form I × A where A ∈ B(R) and I is itself a finite union of intervals (seee.g. [2] for more information about martingale-valued measures). The associated“control measure” is the σ-finite measure µ = λ× ρ, where λ is Lebesgue measureon [0, T ] and ρ is defined on (R,B(R)) by ρ(A) = σ2δ0(A) + ν(A − 0) for allA ∈ B(R). We can easily compute

E(M([0, t]×A)2) = µ([0, t]×A) = tρ(A),

for each t ∈ [0, T ], A ∈ B(R).Returning to the set-up of section 1, we take H = L2(S,B(S), µ;C) so that for

each n ∈ N, H⊗n

= L2(Sn,B(Sn), µn;C) and H⊗n

s comprises symmetric complex-valued square-integrable functions on Sn. Fix n ∈ N and define D(n) to be thelinear space of all functions fn ∈ H⊗n

which take the form

fn =N∑

j1,...,jn=1

aj1,...,jn1Aj1×···×Ajn, (4.4)

where N ∈ N, each aj1,...,jn ∈ C, and is zero whenever two or more of the indicesj1, . . . , jn coincide and A1, . . . , AN ∈ B(S), with Ai of the form Ji × Bi where Ji

is an interval in [0, T ] and Bi ∈ B(R) with ρ(Bi) < ∞, for each 1 ≤ i ≤ N . It isshown as in Proposition 1.6 of Huang and Yan [26] that D(n) is dense in H⊗n

. Itthen follows that D(n)

s is dense in H⊗n

s where D(n)s := D(n) ∩H⊗n

s .For each fn ∈ D(n) we define its multiple Wiener-Levy integral by

In(fn) =N∑

j1,...,jn=1

aj1,...,jnM(Aj1) · · ·M(Ajn). (4.5)


The mapping fn → In(fn) is easily seen to be linear. For each fn ∈ D(n), In(fn) =In(fn), where fn is the symmetrisation of fn.

The next result is due to Ito ([28]). The special case where M is a Brownianmotion is proved in many textbooks (see e.g. [42], [26]) and the general caseproceeds along similar lines.

Theorem 4.4. For each fm ∈ D(m)s , gn ∈ D(n)

s ,m, n ∈ NE(Im(fm)) = 0, E(Im(fm)In(gn)) = n!〈fm, gn〉δmn.

So for each n ∈ N, In is an isometry from D(n)s (equipped with the inner product

〈〈·, ·〉〉 := n!〈·, ·〉) into H. It hence extends to an isometry which is defined on thewhole of H¯n

s . We continue to denote this mapping by In and for each fn ∈ H¯n

s ,we call In(fn) the multiple Wiener-Levy integral of fn. By continuity and Theorem4.4, we obtain

E(Im(fm)) = 0, E(Im(fm)In(gn)) = n!〈fm, gn〉δmn, (4.6)for each fm ∈ H¯m

s , gn ∈ H¯n

s ,m, n ∈ N.

We introduce the n-simplex ∆n in [0, T ] so

∆n = 0 < t1 < · · · < tn < Tand define the iterated stochastic integral

Jn(fn) : =∫

∆n×Rn

fn(w1, . . . , wn)M(dw1) · · ·M(dwn)

=∫ T

0

∫

R

∫ tn−

0

∫

R· · ·

∫ t2−

0

∫

Rfn(t1, x1, . . . , tn, xn)

× M(dt1, dx1) · · ·M(dtn, dxn),

for each fn ∈ H⊗n

. Then if fn ∈ H¯n

s , we have

In(fn) = n!Jn(fn). (4.7)

This is established by exactly the same argument as the case where M is a Brow-nian motion (see e.g. [42]) i.e. first establish the result when fn ∈ D(n)

s by adirect (but messy) calculation and then pass to the general case by means of anapproximation.

The final result in this section is the celebrated chaos decomposition which isagain due to Ito [28].

Theorem 4.5.

H = C⊕∞⊕

n=1

Ran(In). (4.8)

Proof. This follows by iteration of the Ito representation as in Theorem 4 of [38].¤

We use U to denote the mapping⊕∞

n=0 In from Γ(H) to H, where I0(f0) := f0for all f0 ∈ C. U is sometimes called the duality transform. The next result dates

134 DAVID APPLEBAUM

back to Segal [52] in its Gaussian version. The extension to Levy processes firstseems to have been made explicit by Dermoune [14].

Corollary 4.6. [Wiener-Segal-Ito Isomorphism] U is a unitary isomorphism be-tween Γ(H) and H.

Proof. It follows from (4.6) that U is an isometry. By (4.8), if F ∈ H, there existsa sequence (fn, n ∈ N) with each fn ∈ H¯n

s , such that F =∑∞

n=0 In(fn). Thuswe see that U is surjective, and hence is unitary. ¤

4.4. The Role of Stochastic Exponentials. We have seen in section 1 thatexponential vectors play an important structural role in Fock space (see also [37],[41], [46]). In this section we will find the analogous vectors in Levy-Ito space.

Let Y = (Y (t), 0 ≤ t ≤ T ) be a complex valued semimartingale defined on(Ω,F , (Ft, t ≥ 0), P ) so that each Y (t) = Y (1)(t) + iY (2)(t), where Y (1) and Y (2)

are real valued semimartingales. The unique solution to the stochastic differentialequation (SDE)

dZ(t) = Z(t−)dY (t), (4.9)with initial condition Z(0) = 1 (a.s.) is given by the stochastic exponential or

Doleans-Dade exponential,

Z(t)

= expY (t)− Y (0)− 1

2[Y (1)

c , Y (1)c ](t) +

12[Y (2)

c , Y (2)c ](t)− i[Y (1)

c , Y (2)c ](t)

×∏

0≤s≤t

[1 + ∆Y (s)

]e−∆Y (s) (4.10)

for each 0 ≤ t ≤ T . Details can be found in [19] or [29]. Here [·, ·] is the quadraticvariation and Y j

c is the continuous part of Y j for j = 1, 2. Henceforth we willwrite each Z(t) := EY (t).

For each f ∈ H, we introduce the square-integrable martingales (Yf (t), 0 ≤ t ≤T ) defined by

Yf (t) =∫ t

0

f(s, x)M(ds, dx) = σ

∫ t

0

f(s)dB(s) +∫ t

0

∫

R−0f(s, x)N(ds, dx),

where f(s) := f(s, 0), for all 0 ≤ s ≤ T .

Theorem 4.7. For all f ∈ H,

Ue(f) = EYf(T ).

Proof. Iterating the SDE (4.9) as in [19], p. 189 and using (4.7) we obtain for eachf ∈ H,

EYf(T ) = 1 +

∞∑n=1

Jn(f⊗n

)

=∞∑

n=0

1n!In(f⊗

n

).


Now by Corollary 4.6, we have

Uf¯n

= In(f⊗n

) ⇒ U

(f⊗

n

√n!

)=

1n!In(f⊗

n

),

and the result follows on using the strong continuity of U . ¤

Corollary 4.8. The linear span of EYf(T ), f ∈ H is dense in H.

Proof. This follows immediately from Theorem 4.7 and the fact that exponentialvectors are total in Γ(H). ¤

Example 1 Brownian Motion

In this case X(t) = σB(t) for each t ≥ 0 and it is sufficient to take S = [0, T ],so H = L2([0, T ]) and each Yf (t) = σ

∫ t

0f(s)dB(s). Combining Theorem 4.7 with

(4.10) we obtain the well known result:

Ue(f) = exp

σ

∫ T

0

f(s)dB(s)− σ2

2

∫ T

0

|f(s)2|ds,

see e.g. [46], example 19.9, p.130.

Example 2 The Poisson Process

Let (N(t), t ≥ 0) be a Poisson process having intensity λ > 0. In this caseσ = 0 and ν = λδ1. There is then a unique isomorphism V between H = L2(S, µ)and L2([0, T ]) given by (V f)(s, x) = f(s,1)√

λ, for each s ∈ [0, T ], x ∈ R. Writ-

ing W := Γ(V )UΓ(V −1), we find from (4.10) that for all g ∈ L2([0, T ]), takingIg(t) = 1√

λ

∫ t

0g(s)dN(s)−

√λ

∫ t

0g(s)ds = 1√

λ

∑0≤s≤t g(s)∆N(s)−

√λ

∫ t

0g(s)ds,

we obtain

We(g) = exp

−√λ

∫ T

0

g(s)ds

∏

0≤s≤T

(1 +

∆N(s)√λ

g(s))

(c.f. [46], Example 19.11, p.131).

5. Malliavin Calculus in Levy-Ito Space

In this section we will work with operators on the Levy-Ito space H which areunitary transforms (in the sense of the duality transform) of those defined in Γ(H)in section 1, so H = L2(S,B(S), µ;C) where S = [0, T ] × R. We will frequentlyuse the following result. Let H1 and H2 be complex separable Hilbert spaces andlet V and W be unitary isomorphisms between H1 and H2. Let T be a denselydefined closed linear operator on H1 with domain D. Then it is easily verified thatV TW−1 is closed on the dense domain WD.

5.1. The Gradient. We define the gradient D in H by D := (U ⊗ I)∇−U−1 onthe domain UDom(∇−). Then by Theorems 3.4 and 4.5 we see that

Dom(D) =

∞∑n=0

In(fn);∞∑

n=1

nn!||fn||2 <∞.

136 DAVID APPLEBAUM

The spaces UΞ and (U ⊗I)Ξ⊗H are infinite dimensional Sobolev spaces whichare usually denoted D2

1 and D21(H) (respectively) in this context (see e.g. [26]).

For each f ∈ H,ψ ∈ Dom(∇−), Ua(f)ψ = DfUψ where Df := (I ⊗ εf )D is thedirectional gradient operator. For each s = (t, x) ∈ S, the Malliavin derivativeDs is obtained by writing D = (Ds, s ∈ S) and for each ψ ∈ Dom(∇−) we haveU∇−s ψ = DsUψ (µ a.e.)

In particular, if Ef (T ) is the stochastic exponential of f ∈ H then we have

DsEf (T ) = f(s)Ef (T ),

for all s ∈ S except for a set of µ-measure zero (c.f. [15], formula (I.17)) and from(3.10) we see that if ψ =

∑∞n=0 In(fn) ∈ Dom(D), then

Dsψ(·) =∞∑

n=1

nIn−1(fn(·, s)) µ a.e.,

where fn(·, t) is the symmetric function of n−1 variables obtained by fixing s ∈ S.

5.2. The Divergence. We define the divergence δ in H⊗H = L2(S,Σ, µ;H) bythe prescription δ = U∇+(U−1⊗ I) on the domain (U ⊗I)Dom(∇+). Elements ofH¯n ⊗H are sequences (gn+1, n ∈ Z+) where each gn+1 is a measurable functionof n + 1 variables which is symmetric in the first n of these. We denote thesymmetrisation of gn+1 by gn+1 so that for each s1, . . . , sn, sn+1 ∈ S,

gn+1(s1, . . . , sn, sn+1)

=1

n+ 1

(gn+1(s1, . . . , sn, sn+1) +

n∑

j=1

gn+1(s1, . . . , sj−1, sn+1, sj+1, . . . , sn, sj)).

In the following, we will use the standard notation In(gn+1) := (In⊗I)(gn+1) sothe multiple Wiener-Levy integral only acts on the symmetric part of the function.

We then have the following characterisation of Dom(δ).

Theorem 5.1. X =∑∞

n=0 In(gn+1) ∈ Dom(δ) if and only if∑∞n=0(n+ 1)!||gn+1||2 <∞. We then have

δ(X) =∞∑

n=0

In+1(gn+1).

Proof. It is sufficient to take gn+1 = f¯n ⊗ h, then for each s1, . . . , sn, sn+1 ∈ S,

Symm(f¯n

, h)(s1, . . . , sn, sn+1)

=

(√n!

n+ 1

n∑r=0

f⊗n−r ⊗ h⊗ f⊗

r

)(s1, . . . , sn, sn+1)

=

√n!

n+ 1(f(s1) · · · f(sn)h(sn+1)

+n∑

j=1

f(s1) · · · f(sj−1)h(sj)f(sj+1) · · · f(sn)f(sj)

=√

(n+ 1)!gn+1(s1, . . . , sn, sn+1).


The result follows easily from here. ¤

For each f ∈ H we define the directional divergence operator δf on UE byδf := δ (I ⊗ ε†f ), then δfψ = a†(f)U−1ψ for all ψ ∈ UE .

Arguing as in Theorem 3 of [44] (see also Proposition 3.2 of [16]) it follows thatδ(X) is a non-anticipating extension of the Ito integral

∫SX(w)M(dw) which is

defined in the case where X is predictable (see e.g. [2]). We may now rewriteequation (3.11) as the well-known Ito-Skorohod isometry:

E(δ(X)δ(Y )) =∫

S

E(X(s)Y (s))µ(ds) +∫

S

∫

S

E(DtX(s)DsY (t))µ(ds)µ(dt),

(5.1)for all X,Y ∈ D2

1(H) (c.f. Theorem 3.14 in [16]).

5.3. Independence Structure. For each f ∈ H, write f = f1 + f2, where

f1(t, x) :=f(t, 0) if x = 0

0 if x 6= 0

and

f2(t, x) :=

0 if x = 0f(t, x) if x 6= 0

.

We thus obtain a canonical isomorphism between H and H1 ⊕ H2 where H1 =L2([0, T ], λσ) (λσ := σ2λ is rescaled Lebesgue measure) and H2 = L2(E, λ ⊗ ν)where E := [0, T ] × (R − 0). Now suppose that (Ω,F , P ) is of the form (Ω1 ×Ω2,F1 ⊗ F2, P1 × P2). The canonical example of this is called Levy space in [14]and Wiener-Poisson space in [27]. In this set-up Ω1 is the space of continuousfunctions which vanish at zero equipped with Wiener measure P1 on the σ-algebraF1 generated by the cylinder sets. Ω2 is the set of all Z+-valued measures on E

(where Z+ := Z+∪∞). F2 is the smallest σ-algebra of subsets of Ω2 which permitsall evaluations of measures on Borel sets in E to be measurable and P2 is takento be a Poisson measure on (Ω2,F2) with intensity λ× ν.

Let F1,T := σB(s), 0 ≤ s ≤ T and F2,T := σ∫R−0 xN(s, dx), 0 ≤ s ≤ T

so Fi,T are sub-σ-algebras of Fi for i = 1, 2. Applying Corollary 4.6 separately inH1 and H2 we see that the duality transform factorises as U = U1 ⊗ U2 where Ui

is the duality transform between Γ(Hi) and L2(Ωi,Fi,T , Pi) for i = 1, 2. Applyingthis to the tensor decompositions (3.7) and (3.8) We then obtain

D = π1(DB ⊗ I) + π2(I ⊗DN ),

on Dom(DB)⊗Dom(DN ), and

δ = (δB ⊗ I)π∗1 + (I ⊗ δN )π∗2 ,

on Dom(δB)⊕ Dom(δN ). Here DB and δB are the usual gradient and divergenceassociated to Brownian motion (see e.g. [26], [39], [42], [53]) while DN and δN arethe gradient and divergence associated to Poisson random measures (see e.g. [16],[38], [43], [44], [48], [49]).

138 DAVID APPLEBAUM

5.4. Number Operator. The number operator in Levy-Ito space isN = UNU−1 and the corresponding semigroup is (Tt, t ≥ 0) where Tt = UTtU

−1

for each t ≥ 0. We observe that by Theorem 3.6 we have

δD = N ,on Dom(N ) and by Theorem 3.7 we obtain the L2-equivalence of norms wherebythe operator (I +N )−

12 is a unitary isomorphism between H and D2

1.If we employ the independence structure we have

N = NB ⊗ I + I ⊗NN ,

on Dom(NB)⊗Dom(NN ), and

Tt = T Bt ⊗ T N

t ,

for all t ≥ 0, where the sub/superscripts B and N refer to the Brownian andPoisson components in the obvious way.−NB is the well-known infinite-dimensional Ornstein-Uhlenbeck operator which

enjoys the hypercontractivity property. Surgailis [55] has shown that −NP doesnot have this property. Furthermore in Theorem 5.1 of [56], Surgailis proves thatif (Rt, t ≥ 0) is a contraction semigroup in L2(E, λ × µ) then the contractionsemigroup (Rt, t ≥ 0) where each Rt = U2Γ(Rt)U−1

2 is positivity preserving ifand only if (Rt, t ≥ 0) is doubly Markovian. In this latter case, (Rt, t ≥ 0) isitself Markovian. So we can assert the Markovianity of (T N

t , t ≥ 0) and hence of(Tt, t ≥ 0). Further studies of the semigroup (T N

t , t ≥ 0) can be found in [12] and[57].

We finish this article with a word of warning. The universal Malliavin calculusthat we have described here shows great promise for obtaining more widerangingproperties which hold generally for processes which enjoy a chaos decomposition.However there will be local features of the calculus which are particular to theprocess under consideration and which cannot be obtained through the Fock spaceisomorphism. For an example, see formula (I.25) in [15] for an algebraic relationbetween the divergence and the gradient where there is an extra term in the Poissoncase which is absent in Gaussian spaces.

Acknowledgements. This paper grew out of two lecture courses which I gaveat the Universities of Sheffield (2005) and Virginia (2006), respectively. I would liketo thank all the participants in these for their contributions. Particular thanks aredue to Len Scott for arranging the trip to Virginia and for his warm and generoushospitality during my visit. I am also grateful to Martin Lindsay for a helpfuldiscussion about ∇− and ∇+ and to Nick Bingham for drawing my attention to[11]. I would like to thank Fangjun Hsu for reading through an early draft of thisarticle and making some very helpful suggestions. Thanks are also due to bothNick Bingham and Robin Hudson for valuable comments in this regard. I wouldalso like to thank the referee for helpful remarks.

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David Applebaum: Department of Probability & Statistics, University of Sheffield,Hicks Building, Hounsfield Road, Sheffield, England, S3 7RH


SAMPLE PROPERTIES OF RANDOM FIELDSI: SEPARABILITY AND MEASURABILITY

JURGEN POTTHOFF

Abstract. The well-known results about the existence of separable, measur-able resp., modifications of stochastic processes (e.g., [4, 5]) are generalizedto the case of real valued random fields indexed by a separable, separable andlocally convex resp., metric space.

1. Introduction

This is the first in a series of papers in which sample properties of random fieldsare studied. In the present paper the question of existence of modifications of arandom field indexed by a metric space which are separable, measurable resp., isconsidered. In two other papers continuity [6] and — in case that the index set isan open subset of Rm — differentiability [7] are addressed.

From the beginning of general theory of stochastic processes an important ques-tion has been, how statistical properties of a stochastic process determine analyticproperties of its sample paths. The first — and probably most famous — resultin this direction is, of course, the celebrated Kolmogorov-Chentsov theorem, ofwhich a preliminary form by Kolmogorov in 1934 has been reported in a paper bySlutsky [8]. (A quite general form of this theorem is given in [6].) A systematictreatment of this type of questions can be found in the books by Doob [4] and byLoeve [5] (cf. also [2, 1]).

On the other hand, recently there was a growing interest in random fields, forexample within the framework of stochastic partial differential equations. In thepresent series of papers the author generalizes results in [4, 5] to the case where theunderlying index set is a metric space, which seems to be a broad enough settingfor most applications.

Let (Ω,A, P ) be a probability space and let (M,d) be a separable metric space.Throughout this paper we consider real or extended real valued random fields φindexed by M , i.e.,

φ : M × Ω → R or R,(x, ω) 7→ φ(ω, x),

and for every x ∈ M , the mapping ω 7→ φ(x, ω) from Ω into R or R, is B(R)-A-measurable, B(R)-A-measurable respectively. (As it is custom, the second argu-ment of φ is often suppressed.)

2000 Mathematics Subject Classification. 60G60, 60G17.Key words and phrases. Random fields, separability, measurability.

143



144 J. POTTHOFF

The first question addressed in this paper concerns the existence of a modifica-tion of φ which is separable, where separability is defined in analogy with the caseof stochastic processes (cf. [4, 5] and section 2). It turns out that the argumentsin [4, 5] can be generalized in a rather straightforward way, and the result is thatevery random field φ as above admits a separable modification. Moreover, if φ iscontinuous in probability, then every countable dense subset of M is a separatingset.

Assume that (M,d) is equipped with its Borel σ-algebra B(M), and that we aregiven a σ-finite measure µ on (M,B(M)). Similarly as for stochastic processes,a random field is called measurable, if it is measurable as mapping from M × Ω,equipped with the product σ-algebra, to R (or R). It is called a.e. measurable,if it is measurable when restricted to the complement of a µ ⊗ P -null set. Thesecond question considered here is whether a given random field indexed by Mhas a measurable or a.e. measurable modification. This problem necessitates moreserious modifications of the arguments found in [4, 5]. The key is the existence ofan appropriate partition of unity in case that (M,d) is in addition locally compact,cf. [3]. With this additional assumption on (M,d) it is proved in section 3, that thecontinuity in probability of the random field is enough to guarantee the existenceof an a.e. measurable modification.

2. Separability

In this section, we assume throughout that (M,d) is a separable metric space,and φ is a real valued random field indexed by M defined on the probability space(Ω,A, P ). We are interested in the question of existence of a separable modificationof φ. Most of this section carries over from the classical literature, especially from[4] or [5], with only minor modifications. The following definition of separabilityis modelled after the one given in [4] for stochastic processes.

Definition 2.1. A real valued random field φ on (Ω,A, P ) indexed by a metricspace (M,d) is called separable, if there exists an at most countable subset S ofM which is dense in (M,d), so that for all closed intervals C in R, and all opensubsets O of M ,

φ(x) ∈ C, x ∈ O

=φ(x) ∈ C, x ∈ O ∩ S

holds. Then S is called a separating set for φ.

As in [5], separability of φ can be expressed equivalently in various ways:

Lemma 2.2. A real valued random field φ on (Ω,A, P ) indexed by (M,d) is sepa-rable with separating set S, if and only if one of the following equivalent statementsholds:

(S1) For every open subset O in M ,

infy∈O∩S

φ(y) = infx∈O

φ(x), and

supy∈O∩S

φ(y) = supx∈O

φ(x);

SEPARABILITY AND MEASURABILITY OF RANDOM FIELDS 145

(S2) For every open subset O in M ,

infy∈O∩S

φ(y) ≤ infx∈O

φ(x), and

supy∈O∩S

φ(y) ≥ supx∈O

φ(x);

(S3) For every open subset O in M and every x ∈ O,

infy∈O∩S

φ(y) ≤ φ(x) ≤ supy∈O∩S

φ(y);

(S′1) For every x ∈M ,

lim infy→x, y∈S

φ(y) = lim infy→x

φ(y), and

lim supy→x, y∈S

φ(y) = lim supy→x

φ(y);


lim infy→x, y∈S

φ(y) ≤ lim infy→x

φ(y), and

lim supy→x, y∈S

φ(y) ≥ lim supy→x

φ(y);


lim infy→x, y∈S

φ(y) ≤ φ(x) ≤ lim supy→x, y∈S

φ(y).

Proof. The equivalence of statements (S1), (S2), (S3) is obvious. Also the equiva-lence of (S′1), (S′2), (S′3) is clear. Assume that (S2) holds. Let x ∈ M , and choosethe open set O in (S2) as the ball B1/n(x) of radius 1/n, n ∈ N, with center x.Taking the limit n→ +∞, we obtain (S′2). On the other hand, (S′3) implies (S3):Let O be open in M , x ∈ O, and choose n large enough, so that B1/n(x) ⊂ O.Then

infy∈O∩S

φ(y) ≤ infy∈B1/n∩S

φ(y)

≤ supn

infy∈B1/n∩S

φ(y),

and

supy∈O∩S

φ(y) ≥ supy∈B1/n∩S

φ(y)

≥ infn

supy∈B1/n∩S

φ(y).

From (S′3) we have

supn

infy∈B1/n∩S

φ(y) ≤ φ(x) ≤ infn

supy∈B1/n∩S

φ(y),

and therefore (S3) holds. Thus, the equivalence of all statements (Si), (S′i), i =1, 2, 3, has been proven.

146 J. POTTHOFF

Finally we show that the statements (Si), (S′i), i = 1, 2, 3, are equivalent to theseparability of φ with separating set S. To this end assume first that φ is separablewith separating set S. Let ω ∈ Ω, and suppose that O is open in M . Define

a(ω) := infy∈O∩S

φ(y, ω)

b(ω) := supy∈O∩S

φ(y, ω),

where we also allow a(ω) = −∞ or b(ω) = +∞. We set C(ω) := [a(ω), b(ω)] ifa(ω) and b(ω) are finite, and define the closed interval C(ω) in the obvious way inthe case that one of them or both are infinite. Then given ω is such that for ally ∈ O∩S, we have φ(y, ω) ∈ C(ω). Then for ω ∈ Ω we have that for all x ∈ O∩S,φ(x, ω) ∈ C(ω). Because C(ω) is closed, we have

infx∈O

φ(x, ω) ∈ C(ω), and supx∈O

φ(x, ω) ∈ C(ω).

Consequently, (S2) holds. Now suppose that (S1) is true. Given an open set O anda closed interval C = [a, b], let ω ∈ Ω be such that for all y ∈ O ∩ S, φ(y, ω) ∈ C.Then

infx∈O

φ(x, ω) = infy∈O∩S

φ(y, ω)

≥ a.

Similarly, we derive supx∈O φ(x, ω) ≤ b. Therefore we must have φ(x, ω) ∈ C forall x ∈ O, and therefore φ is separable with separating set S. ¤

Lemma 2.3. Let H be a non-empty set, and let ψ be a real valued random field on(Ω,A, P ) indexed by H. Then there exists a non-empty, at most countable subsetS of H, so that for all x ∈ H, and all B ∈ B(R),

P(ψ(y) ∈ B, y ∈ S ∩

ψ(x) /∈ B)= 0.

Corollary 2.4. Let H and ψ be as above, and suppose that (Ck, k ∈ N) is asequence in B(R). Let C ⊂ B(R) be the family of all countable intersections of thefamily (Ck, k ∈ N). Then there exists a non-empty, at most countable subset S ofH, and for every x ∈ H there is a P -null set N(x) so that for every B ∈ C,

ψ(y) ∈ B, y ∈ S ∩

ψ(x) /∈ B ⊂ N(x).

Lemma 2.3 and Corollary 2.4 are proved in [4] (cf. also [5]) for the case that His a subset of R. But it has been remarked in [4], that they hold for a general setH. In fact, the arguments in [4] can be taken over word by word, and thereforethe proofs are omitted here.

Lemma 2.5. Let φ be a real valued random field on (Ω,A, P ) which is indexed byM . Then there exists an at most countable set S ⊂ M , which is dense in (M,d),and for every x ∈ M there is a P -null set N(x) so that for every open subset Oof M , which contains x, and every closed subset C of R,

φ(y) ∈ C, y ∈ O ∩ S ∩

φ(x) /∈ C ⊂ N(x).


Proof. Recall that by hypothesis (M,d) is separable. Let M0 be an at most count-able dense subset of M . We may choose as a countable base of the topology of(M,d) the open balls Br(z) with radius r > 0, r ∈ Q, and centers z ∈ M0. Weapply Corollary 2.4 to the following situation: We choose as the family (Ck, k ∈ N)of Borel sets in R the family of all (bounded or unbounded) closed intervals withrational endpoints. Then the family C is the family of all closed subsets of R.Furthermore, we choose H = Br(z), r > 0, r ∈ Q, z ∈ M0, ψ = φ. As a resultwe obtain a non-empty, at most countable subset Sr,z of Br(z), so that for everyx ∈ Br(z) there is a P -null set Nr,z(x), and the inclusion

φ(y) ∈ C, y ∈ Sr,z

∩ φ(x) /∈ C ⊂ Nr,z(x)

holds for every C ∈ C. Now set

S :=⋃

r>0, r∈Q, z∈M0

Sr,z,

and for x ∈M ,N(x) :=

⋃

r>0, r∈Q, z∈M0

Nr,z(x).

Then S is at most countable, and we have S ∩ Br(z) 6= ∅ for all r > 0, z ∈ M0.Hence S is dense in (M,d). Furthermore, for every x ∈M , P

(N(x)

)= 0.

Next let C ∈ C, x ∈ M , and let O ⊂ M be open with x ∈ O. Then there arer > 0, r ∈ Q, and z ∈M0 with x ∈ Br(z) ⊂ O. Therefore we get

φ(y) ∈ C, y ∈O ∩ S ∩

φ(x) /∈ C

⊂ φ(y) ∈ C, y ∈ Br(z) ∩ S

∩ φ(x) /∈ C

=φ(y) ∈ C, y ∈ Sr,z

∩ φ(x) /∈ C

⊂ Nr,z(x)

⊂ N(x),

and the proof is finished. ¤

Theorem 2.6. Let (M,d) be a separable metric space, and let φ be a real valuedrandom field indexed by M . Then φ has a separable modification.

Proof. Let x ∈M , and let S and N(x) be as in the statement of Lemma 2.5. Letω ∈ N(x). For r > 0, r ∈ Q, and z ∈M0, so that x ∈ Br(z), we set

Cr,z(ω) :=φ(y, ω), y ∈ Br(z) ∩ S

=φ(y, ω), y ∈ Sr,z

,

where A indicates the closure of the set A in R, and Sr,z := Br(z) ∩ S. Byconstruction Cr,z(ω) is closed, and because Sr,z is non-empty, we have that Cr,z(ω)is also non-empty. Moreover, since ω ∈ N(x) is such that for all y ∈ Sr,z the valuesφ(y, ω) belong to Cr,z(ω), Lemma 2.5 entails that φ(x, ω) ∈ Cr,z(ω). Therefore

C(x, ω) :=⋂

r>0, r∈Q, z∈M0, x∈Br(z)

Cr,z(ω)

148 J. POTTHOFF

is closed and φ(x, ω) ∈ C(x, ω). For x ∈ S, ω ∈ Ω or x /∈ S, ω /∈ N(x) set

φ∗(x, ω) := φ(x, ω),

and for x /∈ S, ω ∈ N(x) define

φ∗(x, ω) := lim infy→x, y∈S

φ(y, ω).

It is clear that φ∗ is a modification of φ. Moreover, by construction we have for allω ∈ Ω, x ∈ M that φ′(x, ω) ∈ C(x, ω). We use this to show that φ∗ is separablewith separating set S: Let C be a closed interval, and suppose that O ⊂ M isopen. We have to prove that if ω ∈ Ω is such that φ∗(y, ω) ∈ C for all y ∈ O ∩ S,then φ∗(x, ω) ∈ C for all x ∈ O. First let O = Br(z), r > 0, r ∈ Q, z ∈M0, and letω ∈ Ω be such that φ∗(y, ω) ∈ C for all y ∈ Br(z)∩S = Sr,z. The definition of φ∗

implies that φ(y, ω) ∈ C for all y ∈ Br(z)∩ S = Sr,z. Then by the construction ofC(x, ω) we have that C(x, ω) ⊂ C for all x ∈ Br(z). Since for all (x′, ω′) ∈M ×Ω,φ∗(x′, ω′) ∈ C(x′, ω′) holds, we find φ∗(x, ω) ∈ C. We have shown

φ∗(y) ∈ C, y ∈ Br(z) ∩ S

=

φ∗(x) ∈ C, x ∈ Br(z)

.

Now let O be a general open set. Then O can be written as a (countable) union ofballs of the type Br(z). Therefore, it suffices to take the corresponding intersectionon both sides of the last equality to finish the proof. ¤

Definition 2.7. A real valued random field φ on (Ω,A, P ) which is indexed byM is called a.s. separable, if it is a.s. equal to a separable random field φ∗. If S isthen a separating set for φ∗, it is called an a.s. separating set for φ.

The following two results can be proved as in [4] or [5] without any modification,and therefore the proofs are omitted here.

Lemma 2.8. Assume that φ is a.s. separable with a.s. separating set S. Let M0

be any at most countable dense subset of M , and suppose that for every x ∈ M ,there exists a P -null set N(x), so that one of the properties (S′1), (S′2), or (S′3)holds outside of N(x). Then M0 is a.s. separating for φ.

Theorem 2.9. Let φ be a real valued random field indexed by M, which is contin-uous in probability and is a.s. separable. Then any at most countable dense subsetof M is a.s. separating for φ.

Corollary 2.10. Let φ be a real valued random field indexed by M , which iscontinuous in probability. Then for any at most countable dense subset M0 inM , φ has a modification which is continuous in probability and separable withseparating set M0.

Proof. According to Theorem 2.6, we can choose a modification φ∗ of φ which isseparable for some at most countable dense subset S of M . As a modification ofφ, φ∗ has the same finite dimensional distributions as φ, and therefore also φ∗ iscontinuous in probability. By Theorem 2.9, for any at most countable dense subsetM0 of M φ∗ is a.s. separable. Let NM0 be the exceptional set, and define φ∗∗ asidentically zero on NM0 and as equal to φ∗ on its complement. Then it is obviousthat φ∗∗ is a separable modification of φ which is continuous in probability. ¤


3. Measurability

Throughout this section we assume that (M,d) is a separable, locally compactmetric space. We equip M with its Borel σ-algebra denoted by B(M), and supposethat a σ-finite measure µ is given on (M,B(M)).

Definition 3.1. Let φ be a real valued random field on (Ω,A, P ) indexed by M .(a) φ is called measurable, if the mapping

φ : M × Ω → R

is (B(M)⊗A)-B(R)-measurable.(b) φ is called a.e. measurable (with respect to µ⊗ P ), if there is a µ⊗ P -null

set, so that on its complement φ coincides with a measurable random field.

We investigate in this section the question under which conditions a given realvalued random field φ indexed by M has a measurable modification. To thisend, we combine the arguments given in [4] with the existence of an appropriatepartition of unity (cf., e.g., [3]).

We begin with a lemma which will later on allow us to assume without loss ofgenerality that µ is finite.

Lemma 3.2. There exists a finite measure on (M,B(M)) which is equivalent toµ.

Proof. By hypothesis there exists a sequence (Bn, n ∈ N) in B(M) so that M =⋃nBn, and for every n ∈ N we have µ(Bn) < +∞. For A ∈ B(M) set

µ(A) :=∞∑

n=1

2−n µ(A ∩Bn)1 + µ(Bn)

.

It is straightforward to check that µ is a finite measure on (M,B(M)). Also, itis obvious that µ is absolutely continuous with respect to µ. On the other hand,suppose that A ∈ B(M) is such that µ(A) = 0. Then it follows that µ(A∩Bn) = 0for every n ∈ N. Since M =

⋃nBn, we find that µ(A) = 0, and therefore µ is

absolutely continuous with respect to µ. ¤

Given the random field φ as above, we construct a sequence (φn, n ∈ N) of realvalued random fields indexed by M as follows.

By hypothesis there exists an at most countable subset M0 of M which is densein (M,d). We choose as a base B of the topology of (M,d) the family of open ballswith rational radii and centers in the set M0. Fix n ∈ N. Let C0

n denote the familyof open balls of radius 1/n with centers in M0. Then C0

n is an open covering of M .According to [3, No. 12.6.1], there exists an at most countable finer covering Cn ofM by sets in B, which is locally finite: There exists a sequence (xn,m, m ∈ N) inM0, and a sequence (rn,m, m ∈ N), rn,m > 0, rn,m ∈ Q, so that

Cn = (Bn,m, m ∈ N),

where Bn,m is the ball of radius rn,m with center xn,m. Cn is finer than C0n in

the sense that for every m ∈ N there exists a set C ∈ C0n so that Bn,m ⊂ C.

Consequently, rn,m ≤ 1/n for all m ∈ N. Moreover, for every x ∈ M there is a

150 J. POTTHOFF

neighborhood U of x, so that U ∩ Bn,m = ∅ for almost all m ∈ N. In particular,every x ∈ M belongs only to finitely many balls in Cn. In [3], 12.6.3, it is statedthat there exists a continuous partition of unity (fn,m, m ∈ N) subordinate to Cn:For every m ∈ N, fn,m is a continuous function from M to R, such that for allx ∈M , 0 ≤ fn,m(x) ≤ 1,

∞∑m=1

fn,m(x) = 1,

and supp fn,m ⊂ Bn,m. We define

φn(x) :=∞∑

m=1

φ(xn,m) fn,m(x), x ∈M. (3.1)

It is an easy exercise to show that the random fields (x, ω) 7→ φ(xn,m, ω) fn,m(x)are measurable, and therefore so is φn for every n ∈ N.

Furthermore, if for every x ∈ M we have φ(x) ∈ L1(P ), then for every n ∈ Nand every x ∈ M , φn(x) ∈ L1(P ): In view of equation (3.1) this follows from thefact that for every n ∈ N and every x ∈M there are only finitely many m ∈ N sothat fn,m(x) 6= 0, and that we have |fn,m(x)| ≤ 1.

Lemma 3.3. Suppose that for every x ∈M , φ(x) belongs to L1(P ) and that

φ : M → L1(P )

x 7→ φ(x)

is continuous. Then for every x ∈ M , the sequence (φn(x), n ∈ N) converges inL1(P ) to φ(x).

Proof. We shall write ‖ · ‖1 for the pseudo-norm of L1(P ). Let x ∈ M , ε > 0.Choose δ > 0 so that for all y ∈M , d(x, y) < δ implies ‖φ(x)−φ(y)‖1 < ε. Choosen0 ∈ N with 1/n0 < δ. Let n ∈ N be such that n ≥ n0. Note that for m ∈ N, wehave that fn,m(x) > 0 implies x ∈ Bn,m, i.e., d(x, xn,m) < rn,m ≤ 1/n < δ. Thuswe can estimate as follows

‖φ(x)− φn(x)‖1 =∥∥∥

∞∑m=1

(φ(x)− φ(xn,m)) fn,m(x)∥∥∥

1

≤∞∑

m=1

‖φ(x)− φ(xn,m)‖1 fn,m(x)

<

∞∑m=1

ε fn,m(x)

= ε,

and the proof is finished. ¤

Theorem 3.4. Assume that φ is a real valued random field indexed by M whichis continuous in probability. Then φ has an a.e. measurable modification. Fur-thermore, if in addition φ is separable with separating set S ⊂ M , then the a.e.measurable modification can be chosen in such way that it is separable with sepa-rating set S.


Remark 3.5. If we assume in addition that (M,d) is complete with respect tod, then it becomes a Borel space, and in this case (even without the assumptionof local compactness) the statement of the theorem follows directly from Doob’sclassical results [4, p.60 ff].

Proof. Throughout this proof we use the notation already employed above. With-out loss of generality we may assume that φ is uniformly bounded. (Otherwise,we consider instead of φ the random field arctan φ, construct its modification,and undo the transformation by arctan at the end of the proof.) Also, by Corol-lary 2.10, we may assume that φ is separable with separating set S, where S isany at most countable dense subset of M , and we choose M0 = S in the aboveconstruction of the sequence (φn n ∈ N).

First observe that for every x ∈M we have x ∈ B(M), because B(M) containsall closed sets. This entails that the separating set S belongs to B(M), and hencethe restriction φS of φ to S × Ω is measurable: If B ∈ B(R), then

φ−1S (B) = φ−1(B) ∩ (S × Ω)

=⋃

x∈S

φ−1(B) ∩ (x × Ω)

=⋃

x∈S

x × φ(x)−1(B),

and the sets x × φ(x)−1(B), x ∈ S, belong to B(M) ⊗ A. Since S is at mostcountable, it follows that also their union over x ∈ S is in B(M) ⊗ A. Thereforewe may leave φ on S × Ω unchanged, and it remains to construct the desiredmodification on S × Ω.

Since φ is uniformly bounded, the family (φ(x), x ∈ M) is trivially uniformlyintegrable. Thus the assumption of continuity in probability implies that x 7→ φ(x)is continuous from M into L1(P ). Consider now the sequence (φn, n ∈ N) as inequation 3.1, with M0 = S. By construction, for every n ∈ N, φn is measurable,and by Lemma 3.3 we know that for every x ∈ M , (φn(x), n ∈ N) converges inL1(P ) to φ(x). Because φ is uniformly bounded, we see from equation 3.1 thatso is the sequence (φn, n ∈ N). Moreover, the measure µ is bounded, so that thedominated convergence theorem gives us that

∫

S

‖φ(x)− φn(x)‖1 dµ(x) → 0, n→ +∞.

By an application of Fubini’s theorem, we therefore find that the sequence (φn, n ∈N) is Cauchy in L1(S×Ω,B(S)⊗A, µ⊗P ), where B(S) is the trace of B(M) onS. We abbreviate the latter L1-space with L1(S ×Ω) in the sequel. The Riesz-Fischer-theorem implies that there exists ψ in L1(S × Ω) so that (φn, n ∈ N)converges in L1(S ×Ω) to ψ. In particular, ψ is measurable from S ×Ω into R.Moreover, by selection of a subsequence, we may suppose that there is a µ⊗P -nullset N ∈ B(S) ⊗ A, so that on S × Ω \ N the sequence (φn, n ∈ N) convergespointwise to ψ.

152 J. POTTHOFF

We use again Fubini’s theorem and observe that∫

S

‖φn(x)− ψ(x)‖1 dµ(x) → 0, n→ +∞.

By selection of another subsequence, if necessary, we therefore obtain that there isa µ-null set S0 in B(S), so that for all x in its complement we have φn(x) → ψ(x),n→ +∞, in L1(P ). Since this subsequence converges also to φ(x) we have for allx ∈ S0, P (φ(x) = ψ(x)) = 1.

We now define the modification φ∗ of φ as follows:

φ∗(x, ω) :=

φ(x, ω), (x, ω) ∈ (

(S ∪ S0)× Ω) ∪N

ψ(x, ω), otherwise.

We have already shown above that for all x ∈ M , P(φ∗(x) = φ(x)

)= 1, i.e.,

φ∗ is indeed a modification of φ. Furthermore, φ∗ is measurable when restrictedto S × Ω or to S0 × Ω. Since S0 is a µ-null set, S0 × Ω is a µ ⊗ P -null set, andconsequently φ∗ is a.e. measurable.

Finally we show that φ∗ is separable with separating set S. Let O be open inM , let C be a closed interval, and assume ω ∈ Ω is such that for all y ∈ O ∩ S wehave φ∗(y, ω) ∈ C. By construction, φ∗ and φ coincide on S×Ω, so that we obtainφ(y, ω) ∈ C for all y ∈ O∩S. Let x ∈ O. We have to show that φ∗(x, ω) ∈ C. Thisis trivial for x ∈ S. For (x, ω) ∈ M × Ω so that x ∈ S0 or (x, ω) ∈ N this followsfrom the fact that φ is separable with separating set S. It remains to consider thecase where (x, ω) ∈ M × Ω is such that x ∈ S0 and (x, ω) ∈ S × Ω \ N . Letr > 0 be such that Br(x) ⊂ O. Choose n0 ∈ N so that 1/n0 ≤ r, and considern ∈ N with n ≥ n0. Then by construction of φn(x) in equation (3.1), we have thatthose m ∈ N, for which fn,m(x) > 0, are such that d(x, xn,m) < 1/n ≤ r. Thusxn,m ∈ Br(x) for those terms which contribute to (3.1), and the correspondingvalues of φ(xn,m, ω) are by assumption in C. φn(x, ω) is a convex combination ofthese values, and therefore φn(x, ω) ∈ C, for all n ∈ N, n ≥ n0. Now φ∗(x, ω) isby construction the limit of a subsequence of (φn(x, ω), n ∈ N, n ≥ n0), and C isclosed. Hence we get φ∗(x, ω) ∈ C, and the proof is finished. ¤

Acknowledgement. It is a pleasure to thank H.-P. Butzmann, G. Di Nunno,S. Kruse, W. Seiler and H. Watanabe for stimulating discussions. Moreover, theauthor gratefully acknowledges the helpful comments by an anonymous refereewhich led to Remark 3.5.

References

1. Adler, R.: The Geometry of Random Fields, Wiley, New York, 1981.2. Cramer, H. and Leadbetter, M. R.: Stationary and Related Stochastic Processes, Wiley, New

York, 1967.3. Dieudonne, J: Treatise on Analysis, vol. II, Academic Press, New York, San Francisco,

London, 1970.4. Doob, J. L.: Probability, Wiley, New York, 1953.5. Loeve, M: Probability Theory, 4th ed., vol. II, Springer, New York, Heidelberg, Berlin, 1978.6. Potthoff, J.: Sample properties of random fields II — Continuity, E-print, University of

Mannheim, 2008, http://ls5.math.uni-mannheim.de.


7. Potthoff, J.: Sample properties of random fields III — Differentiability, E-print, Universityof Mannheim, 2008, http://ls5.math.uni-mannheim.de.

8. Slutsky, E. E.: Qualche proposizione relative alla teoria delle funzioni aleatorie, Giorn. Ist.Attuari 8 (1937), 183–199.

Jurgen Potthoff:Lehrstuhl fur Mathematik V, Universitat Mannheim, D–68131Mannheim, Germany


A CLASS OF ANTICIPATING LINEAR STOCHASTICDIFFERENTIAL EQUATIONS

JULIUS ESUNGE

Abstract. In this paper, we present the white noise methods for solvinglinear stochastic differential equations of anticipating type. Such equationsmay be solved using the S-transform, an important tool within the white noisetheory. This approach provides a useful remedy to the fact that the Ito theoryof stochastic integration is inapplicable to such equations. The technique ispresented with several examples, including an application to finance.

1. Introduction

Let B(t) be a Brownian motion and consider the stochastic integral equation

X(t) = X(a) +∫ t

a

f(s,X(s)) dB(s) +∫ t

a

g(s,X(s)) ds, (1.1)

where t ∈ [a, b], a, b ∈ [0,∞). Equation (1.1) is an Ito stochastic integral equationif X(a) is measurable with respect to σB(s) : s ≤ a. This Ito stochastic integralequation has a unique continuous solution provided that f and g satisfy Lipschitzand growth conditions, that is

(i) there exists C1 > 0 such that for any t ∈ [a, b], and x, y ∈ R,

|f(t, x)− f(t, y)|+ |g(t, x)− g(t, y)| ≤ C1|x− y|,and

(ii) there exists C2 such that for any t ∈ [a, b], and x ∈ R,

|f(t, x)|2 + |g(t, x)|2 ≤ C2(1 + x2),

respectively.We note that the existence of a solution is established by applying the Picard

iteration, where X0(t) = X(a) and for n ≥ 1,

Xn(t) = X(a) +∫ t

a

f(s,Xn−1(s))dB(s) +∫ t

a

g(s,Xn−1(s)) ds

and with probability 1, Xn(t) converges to X(t) on [a, b] uniformly. Interest-ingly enough, if we relax the measurability requirement on X(a) (with respect toσB(s) : s ≤ a), f(s,X(a)) may be anticipating. As a result

∫ t

af(s,X(a)) dB(s)

is not an Ito integral, hence X1(t) is undefined as an Ito process. Moreover,

2000 Mathematics Subject Classification. 60H40.Key words and phrases. White noise analysis, S-transform, anticipating stochastic differential

equations.

155



156 JULIUS ESUNGE

Equation (1.1) would no longer be an Ito stochastic integral equation. Questionsemanating from this situation have been considered by several researchers. Inparticular, we mention the works by Buckdahn [1][2], Leon and Protter [8] andNualart and Pardoux [9], some of which employ techniques from the MalliavinCalculus. The notation and terminology used throughout this paper is standardas may be seen for instance in the books [4][6][10] .

2. The White Noise Methods

We give a brief review of the white noise theory from the book [6].

2.1. The white noise space. Let E be a separable Hilbert space with norm| · |0. Let A be a densely defined self-adjoint operator on E, whose eigenvaluesλnn≥1 satisfy the conditions

• 1 < λ1 ≤ λ2 ≤ · · · ,• ∑∞

n=1 λ−2n <∞.

For any p ≥ 0, let Ep be the completion of E with respect to the norm | f |p =|Apf |0. Observe that Ep is a Hilbert space under the norm | · |p, and Ep ⊂ Eq forall p ≥ q.

In fact, by the second condition on the eigenvalues of A, the inclusion mapi : Ep+1 → Ep is a Hilbert-Schmidt operator (see [6] for details).

Next, let E = projective limit of Ep : p ≥ 0 and let E ′ be its dual.The spaceE = ∩p≥0Ep equipped with the topology given by the family | · |pp≥0 of semi-norms is a nuclear space. Consequently E ⊂ E ⊂ E ′ is a Gel’fand triple withcontinuous inclusions:

E ⊂ Eq ⊂ Ep ⊂ E ⊂ E ′p ⊂ E ′q ⊂ E ′, q ≥ p ≥ 0,

having identified E with itself using the Riesz Representation Theorem.Let 〈·, ·〉 denote the duality theorem between E ′ and E . By Minlos theorem,

there is a unique probability measure µ on the Borel subsets of E ′ such that forany f ∈ E , the random variable 〈·, f〉 is normally distributed with mean 0 andvariance | f |20. It follows that µ is uniquely determined by

∫

E′ei〈x,ξ〉 dµ = e−

12 | ξ|20 , ∀ ξ ∈ E . (2.1)

The probability space (E ′, µ) is known as the white noise space. We denote thespace L2(E ′, µ) by (L2), observing that this space consists of all measurable func-tions h : E ′ → C such that ∫

E′|h(x)|2 dµ(x) <∞.

Within this framework, the somewhat ubiquitous white noise tool, known as theS-transform is defined (see [6]). In fact, if ϕ(t) ∈ (L2), then for ξ ∈ Sc,

Sϕ(t)(ξ) =∫

S′ϕ(t)(x+ ξ) dµ(x).

A CLASS OF ANTICIPATING LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS 157

Meantime, if X and Y are generalized functions, their Wick product, denotedX ¦ Y is the unique generalized function such that

S(X ¦ Y ) = (SX)(SY ).

Clearly, an important feature of the S-transform is that much like the Fouriertransform changes convulotions into products, it turns Wick products into ordi-nary products. It is worthnoting that the Wick product plays an intrinsic rolein stochastic integration, especially when one discusses situations involving antic-ipating initial conditions or integrands.

2.2. The Hitsuda-Skorohod integral. Let ∂t ≡ D∂tbe the white noise differ-

ential operator (also known as the Hida differential operator or the annihilationoperator), as defined in [6]. The adjoint of ∂t, denoted by ∂∗t ≡ D∗∂t

, is called thecreation operator.

Starting with the Gel’fand triple S(R) ⊂ L2(R) ⊂ S′(R) and following [6],one obtains the Gel’fand triple (S)β ⊂ (L2) ⊂ (S)∗β . If ϕ : [a, b] → (S)∗β is

Pettis integrable, then the (white noise) integral∫ b

a∂∗t ϕ(t) dt is called the Hitsuda-

Skorohod integral of ϕ, provided∫ b

a∂∗t ϕ(t) dt is a random variable in (L2).

The Hitsuda-Skorohod integral extends the Ito integral to ϕ(t) which may beanticipating. In fact, if ϕ(t) is nonanticipating and

∫ b

a‖ϕ(t)‖20 dt <∞, then

∫ b

a

∂∗t ϕ(t) dt =∫ b

a

ϕ(t) dB(t).

See [5] or [6] for details.

2.3. The white noise approach. In a bid to circumvent the challenges men-tioned in Section 1 above, one possibility is to replace Equation (1.1), with

X(t) = X(a) +∫ t

a

∂∗sf(s,X(s)) ds+∫ t

a

g(s,X(s)) ds, (2.2)

where∫ t

a∂∗sf(s,X(s)) ds is a Hitsuda-Skorohod integral. Equality in (2.2) is as

random variables in the complex Hilbert space (L2) ≡ L2(ξ′, µ).The white noise methods involve using the S-transform to convert Equation

(2.2) into

SX(t)(ξ) = SX(a)(ξ) +∫ t

a

ξ(s)Sf(s,X(s))(ξ) ds+∫ t

a

Sg(s,X(s))(ξ) ds (2.3)

which is an ordinary integral equation for each fixed ξ ∈ Sc. Next assuming Equa-tion (2.3) can be solved for each ξ, a solution to Equation (2.2) would be obtainedby applying the inverse S-transform, provided of course that taking inverse S-transform is possible. This requires that the solution to Equation (2.3) be in therange of the S-transform of an appropriate space.

158 JULIUS ESUNGE

3. Some Examples

We now turn our attention to a few interesting examples previously consideredin turn by Buckdahn [1][2] and Kuo [6], using different techniques. In order toexplain the key ideas, we describe the arguments from [6] for both examples.

Example 3.1. [1][6, page 280] Let us examine

X(t) = sgn(B(1)) +∫ t

0

X(s) dB(s).

Since sgn(B(1)) /∈ σB(s); s ≤ 1, the preceding equation corresponds to

X(t) = sgn(B(1)) +∫ t

0

∂∗sX(s) ds, t ∈ [0, 1]. (3.1)

So Equation (3.1) is not an Ito stochastic integral equation.To solve Equation (3.1), using the approach in section 2, let SX(t) = F (t) and

S[ sgn(B(1))] = G. Then Equation (3.1) becomes (after we take S-transform)

F (t)(ξ) = G(ξ) +∫ t

0

ξ(s)F (s)(ξ) ds,

so that for each ξ ∈ Sc, we have, with t ∈ [0, 1],

F ′(t) = ξ(t)F (t) and F (0) = G(ξ).

Consequently

F (t)(ξ) = G(ξ) eR t0 ξ(s) ds

= G(ξ) e〈1[0,t),ξ〉 (3.2)

= G(ξ)S(: e〈·,1[0,t)〉 :

)(ξ)

= S(sgn(B(1)))(ξ) S(: e〈·,1[0,t)〉 :

)(ξ)

= S[sgn(B(1)) ¦

(: e〈·,1[0,t)〉 :

)](ξ),

whence we have

X(t) = sgn(B(1)) ¦(: e〈·,1[0,t)〉 :

)

= sgn(B(1)) ¦(eB(t)− t

2

).

It remains to show that X(t) ∈ (L2) for all t. To this end, consider

ϕ(t) = sgn(B(1)− t) eB(t)− t2 = sgn(〈·, 1[0,1]〉 − t) e〈·,1[0,t)〉− t

2 ,

since B(t) = 〈·, 1[0,t)〉.


Next, let us determine Sϕ(t). By definition,

Sϕ(t)(ξ) =∫

S′ϕ(t)(x+ ξ) dµ(x)

=∫

S′sgn

(〈x+ ξ, 1[0,1]〉 − t)e〈x+ξ,1[0,t)〉− t

2 dµ(x)

=∫

S′sgn

(〈y + ξ, 1[0,1]〉)e〈1[0,t),ξ〉 dµ(y)

= e〈1[0,t),ξ〉 S (sgn(B(1))(ξ))= F (t)(ξ) (by Equation 3.2)

Recall that F (t) = SX(t), so we see here that SX(t) = Sϕ(t), with ϕ(t) ∈ (L2)and since S is injective, it follows that

X(t) = ϕ(t) = sgn(B(1)− t) eB(t)− t2 .

Example 3.2. [6, page 282] Consider the stochastic integral equation

X(t) = 1 +∫ t

0

∂∗sX(s) ds+∫ t

0

sgn(B(1)− s) eB(s)− s2 ds (3.3)

where 0 ≤ t ≤ 1. We claim that the solution to Equation (3.3) is given byX(t) = eB(t)− t

2 + tϕ(t), where ϕ(t) = sgn(B(1)− t) eB(t)− t2 .

Let F (t) = S(X(t)) and G = S[sgn(B(1))]. Then we recall from Example 3.1that for

ϕ(t) = sgn(B(1)− t) eB(t)− t2

= sgn(〈·, 1[0,1]〉 − t

)e〈·,1[0,t)〉− t

2 ,

we have Sϕ(t)(ξ) = e〈1[0,t),ξ〉 S(sgn(B(1)))(ξ). Therefore applying the S-transformto (3.3), we get for ξ ∈ Sc

S(X(t))(ξ) = 1 +∫ t

0

ξ(s)S(X(s))(ξ) ds+∫ t

0

S(ϕ(s))(ξ) ds

so that

F (t)(ξ) = 1 +∫ t

0

ξ(s)F (s)(ξ) ds+∫ t

0

e〈1[0,s),ξ〉S(sgn(B(1))(ξ) ds

= 1 +∫ t

0


0

G(ξ)e〈1[0,s),ξ〉 ds,

since G = Ssgn(B(1)). So

F (t)(ξ) = 1 +∫ t

0


0

G(ξ)eR s0 ξ(u) du ds. (3.4)

Now for each ξ ∈ Sc, Equation (3.4) implies that F (t) satisfies the ordinary differ-ential equation

F ′(t) = ξ(t)F (t) +G(ξ)eR t0 ξ(s) ds,

F (0) = 1,(3.5)

160 JULIUS ESUNGE

for t ∈ [0, 1]. We now seek a solution for Equation (3.5), which is just a first orderlinear ordinary differential equation. We have

F ′(t)− ξ(t)F (t) = G(ξ)eR t0 ξ(s) ds

with integrating factor e−R t0 ξ(s) ds. Multiplying through by the integrating factor,

Equation (3.5) becomes

e−R t0 ξ(s) dsF ′(t)− ξ(t)F (t)e−

R t0 ξ(s) ds = G(ξ),

which isd

dt

[F (t)e−

R t0 ξ(s) ds

]= G(ξ).

Therefore

F (t)e−R t0 ξ(s) ds =

∫ t

0

G(ξ) ds+K,

where K is a constant. Hence F (t) is given by

F (t) = tG(ξ)eR t0 ξ(s) ds +Ke

R t0 ξ(s) ds,

and since F (0) = 1, we have K = 1. So

F (t) = eR t0 ξ(s) ds1 + tG(ξ) (3.6)

is the solution to Equations (3.5).Next, we recall that

eR t0 ξ(s) ds = e〈1[0,t),ξ〉 = S

(eB(t)− t

2

)(ξ).

Moreover

S(ϕ(t))(ξ) = S(sgn(B(1)− t)eB(t)− t

2

)(ξ) = G(ξ)e

R t0 ξ(s) ds,

which now allows us to rewrite equation (3.6) as

S(X(t))(ξ) = S(eB(t)− t

2

)(ξ) + t S(ϕ(t))(ξ).

It then follows that

X(t) = eB(t)− t2 + t ϕ(t)

= eB(t)− t2 + t sgn(B(1)− t)eB(t)− t

2

= eB(t)− t2 1 + t sgn(B(1)− t).

The preceding examples provide the basis for our first result:

Theorem 3.1. Let X(t) be a stochastic process such that

S(X(t))(ξ) = G(ξ)eR t0 ξ(s) ds,

where ξ ∈ Sc. Then the solution to the stochastic integral equation

Y (t) = 1 +∫ t

0

∂∗sY (s) ds+∫ t

0

X(s) ds (3.7)

for t ∈ [0, 1] is given by Y (t) = eB(t)− t2 + tX(t).


Proof. Let SY (t) = H(t). By hypothesis S(X(t)) = G(ξ)eR t0 ξ(s) ds. Applying the

S-transform to Equation (3.7), we have

S(Y (t))(ξ) = 1 +∫ t

0

ξ(s)S(Y (s))(ξ) ds+∫ t

0

S(X(s))(ξ) ds (3.8)

⇒ H(t)(ξ) = 1 +∫ t

0

ξ(s)H(s)(ξ) ds+∫ t

0

G(ξ) eR s0 ξ(u) duds

So for each ξ ∈ Sc, we have

H ′(t) = ξ(t)H(t) +G(ξ) eR t0 ξ(u) du, H(0) = 1

⇒ d

dt

[H(t) e−

R t0 ξ(u) du

]= G(ξ)

⇒ H(t) e−R t0 ξ(u) du = tG(ξ) +K

⇒ H(t) = tG(ξ) eR t0 ξ(u) du +K e

R t0 ξ(u) du

Since H(0) = 1 and K = 1, we have

H(t) = eR t0 ξ(u) du + tG(ξ) e

R t0 ξ(u) du,

namely

S(Y (t))(ξ) = S(eB(t)− t

2

)(ξ) + t S(X(t))(ξ),

which implies Y (t) = eB(t)− t2 + tX(t), as desired. ¤

4. A Class of Linear Equations

We now consider a class of equations based on the general linear stochasticintegral equation of Hitsuda-Skorohod type, namely

X(t) = ϕ+∫ t

a

∂∗s (f(s)X(s)) ds+∫ t

a

[g(s)X(s) + ψ(s)] ds,

where f , g are deterministic, ϕ is a random variable and ψ is a stochastic process.We will see how the S-transform can be used to solve this equation. We start byrecalling a lemma (Lemma 13.32 in [6]) and a theorem (Theorem 13.33 in [6]) andclose with another unifying result.

Lemma 4.1. [6] If f ∈ L2([a, b]) and ϕ ∈ Lp(S′) for some p > 2, then

ϕ ¦ e[R t

af(s) dB(s)− 1

2

R ta

f(s)2 ds] =(T−1[a,t]fϕ

)e[R t

af(s) dB(s)− 1

2

R ta

f(s)2 ds],

where ¦ is the Wick product and Thϕ(x) = ϕ(x+ h).

Next we consider a result that shows the solution of the general Hitsuda-Skorohod type stochastic integral equation when certain conditions are specified.

Theorem 4.2. [6] Suppose f(t), g(t) are deterministic functions, ϕ is a randomvariable, and ψ(t) is a stochastic process satisfying

1. f, g ∈ L2([a, b])).2. ϕ ∈ Lp(S′) for some p > 2.3. ψ ∈ Lq([a, b]× S′) for some q > 2.

162 JULIUS ESUNGE

Then the stochastic integral equation

X(t) = ϕ+∫ t

a

∂∗s (f(s)X(s)) ds+∫ t

a

(g(s)X(s) + ψ(s)) ds

has a unique solution in L2([a, b], (L2)) given by

X(t) =(T−1[a,t)fϕ

)e[R t

af(s) dB(s)+

R ta(g(s)− 1

2 f(s)2)ds]

+∫ t

a

(T−1[s,t)fψ(s)

)e[R t

sf(r) dB(r)+

R ts (g(r)− 1

2 f(r)2) dr] ds.

Finally, for simplicity, let

Ef (t) = e[R t0 f(s) dB(s)− 1

2

R t0 f(s)2 ds].

In view of the general linear Hitsuda-Skorohod type stochastic integral equationand the concluding result in the previous section, we have

Theorem 4.3. If f ∈ L2([0, 1]) and ϕ is a random variable, with ϕ ∈ Lp(S′) forsome p > 2. Then the stochastic integral equation

Y (t) = 1 +∫ t

0

∂∗s (f(s)Y (s)) ds+∫ t

0

(ϕ ¦ Ef (s)) ds (4.1)

has a solution given by

Y (t) = Ef (t) + t(ϕ ¦ Ef (t)), t ∈ [0, 1].

Proof. For simplicity, let X(t) = ϕ ¦ Ef (t), and let SY (t) = F (t), Sϕ = G. Sincef ∈ L2([0, 1]), we have

S(Ef (t))(ξ) = eR t0 ξ(s)f(s) ds.

Therefore, by taking the S-transform of Equation (4.1), we have for ξ ∈ Sc,

F (t)(ξ) = 1 +∫ t

0

ξ(s)f(s)F (s)(ξ)ds+∫ t

0

G(ξ) eR s0 ξ(u)f(u) du ds,

so that for each fixed ξ ∈ Sc,

F ′(t) = ξ(t)f(t)F (t)(ξ) +G(ξ) eR t0 ξ(u)f(u) du, with F (0) = 1.

For each fixed ξ ∈ Sc, we haved

dt

[e−R t0 ξ(s)f(s) dsF (t)

]= G(ξ)

which yieldse−R t0 ξ(u)f(u) duF (t) = tG(ξ) +K,

where K is a constant. Hence we have

F (t) = K eR t0 ξ(u)f(u) du + tG(ξ) e

R t0 ξ(u)f(u) du.

Since F (0) = 1, we get K = 1 and thus

F (t) = eR t0 ξ(s)f(s) ds + tG(ξ) e

R t0 ξ(s)f(s) ds.

The conclusion follows once we take the inverse S-transform, that is

Y (t) = Ef (t) + t(ϕ ¦ Ef (t)).

¤


8ApplicationOne situation where anticipating initial conditions and integrands arise is in

the pricing of bonds. In [11], the authors propose a framework for determiningprice dynamics of a bond P (t, T ) at time t, which matures at time T with a fixedexpiration value P (T, T ) = 1 almost surely. In effect, this involves considering astochastic process driven by B(t) which reaches a fixed value at a future time Talmost surely.

The main idea involves considering a stochastic integral equation of the form

Xt = XT −∫ T

t

f(s)X(s)dB(s)−∫ T

t

g(s)X(s)ds (4.2)

where f(s) is anticipating. As the authors point out, one must note that even iff(s) and g(s) are adapted, the terminal condition XT = 1 makes for an anticipat-ing equation. By introducing a time reversal operator (see [11]), Equation (4.2)becomes

Xt = X0 −∫ t

0

f(s)X(s)dB(s)−∫ t

0

g(s)X(s)ds (4.3)

with x(0) = P (T, T ) = x0.Considering Equation (4.3), the authors assert the existence of a unique solution

given by

x(t) = x0Ef (t)exp ∫ t

0

g(s)ds.We will now derive this solution using the white noise methods discussed in this

paper. Indeed, in view of Equation (4.2), let SX(t) = F (t), G = Sx0.Then upon taking S-transforms, we have

F (t)(ξ) = G(ξ) +∫ u

0

[ξ(s)f(s) + g(s)]F (s)(ξ)ds

almost surely, for all u ∈ [0, t]. Using similar arguments as in Theorem 4.2, let

Hξ(u) = G(ξ) +∫ u

0

v(s)F (s)(ξ)ds

where v(u) = ξ(u)f(u) + g(u), so that u-a.s., Hξ(u) = F (u)(ξ),∀ξ ∈ SC , andH ′

ξ(u) = v(u)Hξ(u), u-a.s. in [0, t] with Hξ(0) = G(ξ). Therefore,

H ′ξ(u)− v(u)Hξ(u) = 0

d

duHξ(u)e−

R u0 v(s)ds = 0.

Since Hξ(0) = G(ξ), R = G(ξ) and so Hξ(u) = G(ξ)eR u0 v(s)ds.

F (u)(ξ) = G(ξ)eR u0 v(s)ds,

u-a.e. on [0, t]. Next, since f ∈ L2([0, t]),

S(Ef (u)) = eR u0 ξ(s)f(s)ds.

Consequently, we have

S(x0Ef (u) = (Sx0)(ξ)[SEf (u)(ξ)] = G(ξ)eR u0 ξ(s)f(s)ds.

164 JULIUS ESUNGE

F (u)(ξ) = G(ξ)eR u0 ξ(s)f(s)ds · e

R u0 g(s)ds = G(ξ)Sεf (u)(ξ)e

R u0 g(s)ds.

It follows upon inverting that

X(t) = x0Ef (u)exp ∫ u

0

g(s)ds,

as desired.

Acknowledgment. The author is thankful to Professor Hui-Hsiung Kuo forguiding the development of this paper. Gratitude also go to the anonymous refereewhose kind comments influenced the final version of this paper, and to the LSUVIGRE Committee for generous financial support.

References

1. Buckdahn, R.: Skorohod’s integral and linear stochastic differential equations, Preprint1843, Humbolt Universitt, Berlin, (1988).

2. Buckdahn, R.: Anticipating linear stochastic differential equations, Lecture Notes in Controland Information Sciences 136 (1989) 18-23, Springer Verlag.

3. Deck, T., Potthoff, J., and Vage, G.: A review of white noise analysis from a probabilisticstandpoint, Acta Applicandae Mathematicae 48 (1997) 91-112.

4. Hida, T., Kuo, H.-H., Potthoff, J., and Streit, L.: White Noise: An Infinite DimensionalCalculus, Kluwer Academic Publishers, 1993.

5. Kubo, I. and Takenaka, S.: Calculus on Gaussian White Noise III, in: Proc. Japan Acad.57A (1981) 433-437.

6. Kuo, H.-H.: White Noise Distribution Theory, CRC Press, 1996.7. Kuo, H.-H.: Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics 463 (1975),

Springer Verlag.8. Leon, J. A. and Protter, P.: Some formulas for anticipative Girsanov transformations, in:

Chaos Expansions, Multiple Wiener-Ito integrals and Their Applications, C. Houdre and V.Perez-Abreu (eds.), CRC Press, 1994.

9. Nualart, D. and Pardoux, E.: Stochastic calculus with anticipating integrands, Probab. Th.Rel. Fields 76 (1987) 15-49.

10. Obata, N.: White Noise Calculus and Fock Space, Lecture Notes in Mathematics 1577,Springer Verlag, 1994.

11. Platen, E. and Rebolledo, R.: Pricing via anticipative stochastic calculus, Advances in Ap-plied Probability 26 (1994) 1006-1021.

Julius Esunge: Department of Mathematics, Louisiana State University, BatonRouge, LA 70803, USA


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