FEYNMAN-KAC FORMULA FOR LEVY PROCESSES ´ WITH DISCONTINUOUS KILLING RATE

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2015

FEYNMAN-KAC FORMULA FOR LEVY PROCESSES

WITH DISCONTINUOUS KILLING RATE

KATHRIN GLAU

Abstract. The challenge to fruitfully merge state-of-the-art tech-niques from mathematical finance and numerical analysis has in-spired researchers to develop fast deterministic option pricing meth-ods. As a result, highly efficient algorithms to compute optionprices in Levy models by solving partial integro differential equa-tions have been developed. In order to provide a solid mathe-matical foundation for these methods, we derive a Feynman-Kacrepresentation of variational solutions to partial integro differentialequations that characterize conditional expectations of functionalsof killed time-inhomogeneous Levy processes. We allow for a widerange of underlying stochastic processes, comprising processes withBrownian part, and a broad class of pure jump processes such asgeneralized hyperbolic, multivariate normal inverse Gaussian, tem-pered stable, and -semi stable Levy processes. By virtue of ourmild regularity assumptions as to the killing rate and the initialcondition of the partial differential equation, our results providea rigorous basis for numerous applications, not only in financialmathematics but also in probability theory and relativistic quan-tum mechanics.

Time-inhomogeneous Levy process, killing rate, Feynman-Kac representation, weaksolution, variational solution, parabolic evolution equation, partial integro differ-ential equation, pseudo differential equation, nonlocal operator, fractional Laplaceoperator, Sobolev-Slobodeckii spaces, option pricing, Laplace transform of occupa-tion time, relativistic Schrodinger equation[2000] 35S10, 60G51, 60-08, 47G20, 47G30

1. Introduction

Feynman-Kac formulas play a distinguished role in probability theory and func-tional analysis. Ever since their birth in 1949, Feynman-Kac type formulas havebeen a constant source of fascinating insights in a wide range of disciplines. Theyoriginate in the description of particle diffusion by connecting Schrodingers equa-tion and the heat equation to the Brownian motion, see Kac (1949). A type ofFeynman-Kac formula also figures at the beginning of modern mathematical fi-nance: In their seminal article of 1973, Black and Scholes derived their NobelPrize-winning option pricing formula by expressing the price as a solution to apartial differential equation, thereby rediscovering Feynman and Kacs deep linkbetween the heat equation and the Brownian motion.

Date: February 27, 2015Technische Universitat Munchen, Center for [email protected].

1

http://arxiv.org/abs/1502.07531v1

2 K. GLAU

The fundamental contribution of Feynman-Kac formulas is to link stochasticprocesses to solutions of deterministic partial differential equations. Thus theyestablish a connection between different disciplines that have evolved separately.Although both enjoy great success, transfer between them has remained only in-cidental. This may very well be the reason for applications of Feynman-Kac stillappearing so surprisingly fresh. In computational finance, they enable the develop-ment of option pricing methods by solving deterministic evolution equations. Thesehave proven to be highly efficient, particularly when compared to Monte Carlo sim-ulation. Thus, like other deterministic methods, they come into play wheneverefficiency is essential and the complexity of the pricing problem is not too high.This is the case for recurring tasks such as calibration and real-time pricing andhas given rise to extensive research in computing option prices by solving partialdifferential equations over the last decades. The challenge to extend these meth-ods to price options in advanced jump models has inspired researchers to develophighly efficient and widely applicable algorithms in recent years, see for instanceCont and Voltchkova (2005b), Hilber, Reich, Schwab and Winter (2009), Hilber,Reichmann, Schwab and Winter (2013), Salmi, Toivanen and Sydow (2014) andItkin (2015).

In this article we derive a Feynman-Kac formula so as to provide a solid math-ematical basis for fast option pricing in time-inhomogeneous Levy models usingpartial integro differential equations. While large parts of the literature focus onnumerical aspects of these pricing methods, only little is known about the preciselink between the related deterministic equations and the corresponding conditionalexpectations representing option prices. Our main question therefore is: Underwhich conditions is there a Feynman-Kac formula linking option prices given byconditional expectations with solutions to evolution problems?

In order to further specify the problem, we focus on time-inhomogeneous Levymodels and options whose path dependency may be expressed by a possibly discon-tinuous killing rate. In this setting with A = (A t)[0,T ] the Kolmogorov operator of

a time-inhomogeneous Levy process, killing rate (or potential) : [0, T ]Rd R,source f : [0, T ] Rd R and initial condition g : Rd R, the Kolmogorovequation is of the form

tu+A Ttu+ Ttu = f,

u(0) = g .(1)

Proceeding in an unsophisticated manner, one would typically assume that equa-tion (1) has a classical solution. If the solution u is sufficiently regular to allow foran application of Itos formula and moreover satisfies an appropriate integrabilitycondition, it is straightforward to derive the following Feynman-Kac representation

u(T t, Lt) = E(g(LT ) e

Tth(Lh) dh +

T

t

f(Ts, Ls) e

sth(Lh) dh ds

Ft). (2)

Then, the conditional expectation can be obtained by solving the equation by meansof a deterministic numerical scheme. Such an argumentation hinges on a strongregularity assumption on the solution u and thus implicitly on the data of theequation, g, f , A and . This constitutes a serious restriction on the applicabilityof such an approach.

We, however, pay special attention to identifying conditions for the validity of (2)that are appropriate for financial applications. Here, the choice of killing rates asindicator functions turns out to be the key to a variety of applications, not only inmathematical finance, but also in probability theory as we will show in section 2.The fundamental role of killing rates of indicator type is killing the process outside

FEYNMAN-KAC FORMULA FOR LEVY PROCESSES 3

a specified domain which relates them to occupation times. Moreover, exit timesof stochastic processes may conveniently be approximated by means of killing ratesof the indicator type. Thus discontinuous killing rates form a common root ofexit probabilities, the distribution of the supremum process and prices of path-dependent options such as those of barrier, lookback, and American type. For acharacterization of prices of barrier options and the distribution function of thepast supremum of time-inhomogeneous Levy processes obtained with the help ofkilling rates see Glau (2010) and Glau (2015). For these reasons we will allow fornon-smooth and even discontinuous killing rates in Kolmogorov equation (1).

Discontinuities in the killing rate result in non-smoothness of the solution uof Kolmogorov equation (1). In particular, one cannot expect u C1,2. Assumeu(0) 6= 0 and = 1(,0)d in (1), then x 7 u(t, x) C

2 implies x 7 (x)u(t, x) C, which obviously is a contradiction. Hence, for our purposes, the assumptionItos formula can be applied to the solution u is fruitless. Neither is it reasonable topresume that equation (1) has a classical solution. Let us not only emphasize thatsuch irregularity is inherent in equation (1) if the killing rate is discontinuous, butalso that it is a typical feature of Kolmogorov equations for path-dependent optionprices. Prominent examples are boundary value problems related to barrier optionsin Levy models as well as free boundary value problems for American option prices.In all of these cases, the use of a generalized solution concept is necessary and doesequation (1) allow for a weak, also called variational, formulation for a wide classof stochastic processes.

We prefer weak formulations to other general solution concepts such as viscositysolutions for the following reasons. First, weak formulations are sufficiently gen-eral to apply to pricing equations of most of the relevant option and model types.Second, given existence and uniqueness of a weak solution, it only depends on itsregularity properties whether it is also a viscosity solution or a classical solution.The main reason for our choice, however, is that weak formulations are the theo-retical foundation of Galerkin methods, a rich class of versatile numerical methodsto solve partial differential equations. Galerkin methods rely on an elegant prob-lem formulation in Hilbert spaces that by its very construction leads to convergentschemes as well as to a lucid error analysis. They furthermore distinguish them-selves by their enormous flexibility towards problem types as well as compressiontechniques. Both theory and implementation of Galerkin methods have experienceda tremendous advancement over the past fifty years. They have become indispens-able for todays technological developments in such diverse areas as aeronautical,biomechanical, and automotive engineering.

In mathematical finance, Galerkin pricing algorithms have already been devel-oped even for basket options in jump models. Furthermore, numerical experimentsand error estimates have confirmed their efficiency both in theory as well as inpractice. See Hilber et al. (2013), and e.g. Matache, von Petersdorff and Schwab(2004), Matache, Schwab and Wihler (2005), von Petersdorff and Schwab (2004).Beyond space discretizations, the freedom in the choice of the approximating fi-nite dimensional function space is exploited in Galerkin based model reductiontechniques that have a great potential in financial applications, see Cont, Lantosand Pironneau (2011), Pironneau (2011), and Sachs and Schu (2013), Haasdonk,Salomon and Wohlmuth (2012) and Haasdonk et al. (2012b). Hence, our specificquestion is: Under which conditions on the time-inhomogeneous Levy process L, thekilling rate , the source f and initial condition g is there a unique weak solutionof Kolmogorov equation (1) that allows for a stochastic representation of form (2)?

Feynman-Kac representations for viscosity solutions with application to optionpricing in Levy models have been derived in Cont and Voltchkova (2005a) and

4 K. GLAU

Cont and Voltchkova (2005b). Results linking jump processes with Brownian partto variational solutions had already been proven earlier in Bensoussan and Lions(1982). However, in order to cover some of the most relevant financial models, wehave to consider pure jump processes, i.e. processes without a Brownian component,as well. Pure jump Levy models are able to fit market data with high accuracy, seee.g. Eberlein (2001) and, as a consequence, see Schoutens (2003), Cont and Tankov(2004). Moreover, statistical analysis of high-frequency data supports the choice ofpure jump models, and At-Sahalia and Jacod (2014).

Pure jump processes differ significantly from processes with a Brownian part. Forexample, almost surely every path of a Levy process is of finite quadratic variationonly in the presence of a Brownian component. The Brownian component trans-lates to a second order derivative in the Kolmogorov operator, while the pure jumppart corresponds to an integro differential operator of lower order of differentiation.Accordingly, the second order derivative is only present in Kolmogorov operatorsof processes with a Brownian component. As a consequence, the solution to theKolmogorov equation of a pure jump Levy process does not lie in the Sobolev spaceH1 that is the space of quadratic integrable functions with a square integrable weakderivative. Therefore a more general solution space needs to be chosen. In orderto make an appropriate choice, recall that Levy processes are nicely characterizedthrough the Levy-Khinchine formula by the Fourier transform of their distributionrespectively the symbol. Moreover, the symbol is typically available in terms ofan explicit parametric function and as such is the key quantity to parametric Levymodels. For a wide range of processes, the asymptotic behaviour of the symbol en-sures that the solution of the Kolmogorov equation belongs to a Sobolev-Slobodeckiispace, i.e. it has a derivative of fractional order, see e.g. Hilber et al. (2013). Soas to allow for a wide range of initial conditions, such as the payoff function of acall option in logarithmic variables and the Heaviside step function that relates todistribution functions, we use exponentially weighted Sobolev-Slobodeckii spacesas in Eberlein and Glau (2014).

In order to present the main result of the present article, we introduce the un-derlying stochastic processes, present the Kolmogorov equation with killing rate,its weak formulation as well as the solution spaces of our choice. We denote byC0 (R

d) the set of smooth real-valued functions with compact support in Rd andlet

F() :=

Rd

ei,x (x) dx (3)

the Fourier transform of C0 (Rd) and let F1 its inverse.

Since Levy models typically lead to a considerably better fit to the time-evolutionof financial data when time-dependent parameters are chosen, we base our analysison the class of time-inhomogeneous Levy processes. Let us be given a stochasticbasis (,F , (Ft)0tT , P ) and let L be an R

d-valued time-inhomogeneous Levyprocess with characteristics (bt, t, Ft;h)t0. I.e. L has independent incrementsand for fixed t 0 its characteristic function is

E ei,Lt = e

t0As(i) ds , (4)

where the symbol of the process At for any fixed t 0 equals

At() :=1

2, t+ i, bt

Rd

(ei,y 1 + i, h(y)

)Ft(dy). (5)

Here, for every s > 0, s is a symmetric, positive semi-definite dd-matrix, bs Rd,

and Fs is a Levy measure i.e. a positive Borel measure on Rd with Fs({0}) = 0

andRd

(|x|2 1)Fs(dx) < , and h is a truncation function i.e. h : Rd R such


that{|x|>1} h(x)Ft(dx) < with h(x) = x in a neighbourhood of 0. The maps

s 7 s, s 7 bs and s 7(|x|2 1)Fs(dx) are Borel-measurable with

T

0

(|bs|+ sM(dd) +

Rd

(|x|2 1)Fs(dx))ds < (6)

for every T > 0, where M(dd) is a norm on the vector space formed by thed d-matrices.The Kolmogorov operator of the process L is given by

A t(x) :=1

2

d

j,k=1

j,kt2

xjxk(x)

d

j=1

bjt

xj(x)

Rd

((x+ y) (x)

d

j=1

xj(x)hj(y)

)Ft(dy)

(7)

for every C0 (Rd), where hj denotes the j-th component of the truncation

function h. An elementary calculation shows

A t = F1(AtF()) for all C

0 (R

d), (8)

hence Kolmogorov operator A is pseudo differential operator with symbol A.In order to give the weak formulation of Kolmogorov equation (1) let L2

(0, T ;H

)

denote the space of weakly measurable functions u : [0, T ] H with T0u(t)2H dt 0, G 0 such that

uniformly for all t [0, T ], all R Rd, and all u, v H/2 (Rd),

at(u, v) Cu

H/2 (Rd)

vH

/2 (Rd)

(Continuity (Cont-a))

at(u, u) Gu2

H/2 (Rd)

Gu2L2(Rd). (Garding inequality (Gard-a))

As highlighted in equation in (8), the Kolmogorov operator of a time-inhomo-geneous Levy process is a pseudo differential operator. Its symbol is explicitlyknown for various classes and in general is characterized by the exponent of theLevy-Khinchine representation. Thus, it is interesting express parabolicity of thebilinear form in terms of the symbol. This characterization will allow us in section3 to show parabolicity of the related operator for number of classes of processes.The characterization, moreover is one of the key steps in our proof of Feynman-Kacrepresentation (19).

An extension of the bilinear form to weighted Sobolev-Slobodeckii spaces corre-sponds to a shift of the symbol in the complex plane and we introduce the appro-priate notion for the symbol.

Definition 1.2. We say the symbol A = (At)t[0,T ] has Sobolev index uniformlyin time and weight from [0, T ]R, if t 7 At( + i) is cadlag for each R andthere exist constants, 0 < , C,G1 > 0, G2 0 such that uniformly for allt [0, T ], all R and for all Rd,

At( + i) C

(1 + ||

)(Continuity condition (Cont-A))

(At( + i)

) G1

(1 + ||

)G2

(1 + ||

)(Garding condition (Gard-A))

If A is the symbol of a time-inhomogeneous Levy process, we also say L has Sobolevindex uniformly in time and weight [0, T ]R.

For weight = (1, . . . , d) let

U :={z Cd

(zj) {0} sgn(j)[0, |j |) for j = 1, . . . , d}, (14)

R := sgn(1)[0, |1|] sgn(d)[0, |d|]. (15)

The assertion as well as its proof of the next theorem are a straightforwardgeneralization of Theorem 3.2 in (Glau, 2014). In order to provide a self-containedpresentation the proof is given in Appendix B.

Theorem 1.3. Let the symbol A = (At)t[0,T ] of pseudo differential operator A =(A t)t[0,T ] have for each t [0, T ] a continuous extension on U that is analytic


in the interior

U and satisfies for each t [0, T ]

At(z) C(t)

(1 + |z|

)m(t)for all z U (16)

for some constant C(t),m(t) > 0. Then the following assertions are equivalent.

(i) The operator A is parabolic in H (Rd), L2(R

d) uniformly in time andweight [0, T ]R.

(ii) The symbol A has Sobolev index 2 uniformly in time and weight [0, T ]R.

For a time-inhomogeneous Levy process L and Rd, Theorem 25.17 in Sato(1999) implies that

|x|>1

e,x Ft(dx) < for every 0 t T is equivalent to the

Exponential Moment condition

E[e,Lt] < for every 0 t T (EM())

and for such we have

E[ei+,Lt

]= e

t0As(+i) ds for all Rd and all t 0. (17)

Moreover, Lemma 2.1 (c) in Eberlein and Glau (2014) shows that if

EM() holds for every R, (EM(R))

then for every 0 t T the map z 7 At(z) has a continuous extension to the

domain U which is analytic in the interior

U. Moreover, Theorem 25.17 inSato (1999) and Lemma A.1 show that inequality (16) is satisfied with m(t) = 2for some constant C(t) > 0 for each t [0, T ]. We obtain the following

Corollary 1.4. Let L a time-inhomogeneous Levy process with EM(R). Thenthe the following assertions are equivalent.

(i) The Kolmogorov operator of L is parabolic in H (Rd), L2(R

d) uniformlyin time and weight [0, T ]R.

(ii) L has Sobolev index 2 uniformly in time and weight [0, T ]R.

For an integrable or nonnegative random variable X we denote

Et,x(X) := E(X |Lt = x), for t > 0, E0,x(X) := Ex(X) (18)

where x 7 E(X |Lt = x) is the factorization of the conditional expectation E(X |Lt)and denoting by Ex the expectation w.r.t. Px which is a probability measure suchthat Px(L0 = x) = 1.

Theorem 1.5. Let L an Rd-valued time-inhomogeneous Levy process with EM(R)and with Sobolev index 2 uniformly in time and weight [0, T ]R. Then

(i) for : [0, T ]Rd R measurable and bounded, f L2(0, T ; (H (R

d)))

and g L2(Rd) Kolmogorov equation (1) has a unique weak solution u

W 1(0, T ;H (R

d), L2(Rd));

(ii) if additionally, f L2(0, T ;H l(R

d))for some l 0 with l > (d 2)/2

then for every t [0, T ] and a.e. x Rd

u(T t, x) = Et,x

(g(LT ) e

Tt

h(Lh) dh

+

T

t

f(T s, Ls) e

sth(Lh) dh ds

).

(19)

8 K. GLAU

The assertion of part (i) of the theorem directly follows from Corollary 1.4 and theclassical result on existence and uniqueness of weak solutions to parabolic equations,see e.g. Theorem 23.A in Zeidler (1990). Section 5 is dedicated to the proof of part(ii) of Theorem 1.5.

As an immediate consequence of Theorem 1.5 and Sobolev embedding resultTheorem 8.2 in Nezza, Palatucci and Valdinoci (2011) we obtain

Corollary 1.6. Under the assumptions and notations of Theorem 1.5 in the uni-variate case, i.e. for d = 1, for (1/2, 1] and any fixed t (0, T ), the functionx 7 u(t, x) is -Holder continuous with = 212 , i.e.

supx,yR,x 6=y

|u(t, x) u(t, y)|

|x y|< .

In particular, x 7 u(t, x) is continuous and equality (19) in Theorem 1.5 holds forevery x R.

The rest article is organized as follows: In the next section we outline variousapplications of Feynman-Kac Theorem 1.5 and in the third section we present ex-amples of stochastic processes that satisfy the assumptions of the theorem. Wededicate section 4 to a robustness result for weak solutions which is required in ourproof of Theorem 1.5 that we present in section 5. The section also contains regu-larity result Lemma 5.1 for the solutions to the Kolmogorov equation. A providesessential properties of the symbol and the operator, and B concludes with the proofof Theorem 1.3.

2. Applications

We choose examples from different fields such as physics, probability theory andfinance to illustrate the interdisciplinary benefit of Theorem 1.5. We present eachof the examples in a self-consistent way to facilitate its usage on the one side andthe readability on the other.

Remark 2.1. In several applications there does not exist an Rd such that theinitial condition g L2(R

d). In these cases, g may be split up in 2d summands that

are supported in the 2d orthants. By the linearity of the expectation, respectivelyof the PIDE, the problem can be split additively in 2d separate problems. Thentypically for each of the summands gj for j = 1, . . . , d, an exponential dampingfactor j Rd exists such that gj L2j (R

d) and the results of Theorem 1.5 can be

applied for each initial condition gj separately. Consider e.g. the initial conditiong 1. In the one-dimensional setting, write e.g. g = 1(,0] + 1(0,), where

1(,0] L2(R) for every

> 0 and 1(0,) L2+(R) for every

+ < 0.

For numerical purposes, a splitting in smooth functions instead of Heaviside stepfunctions is preferable.

2.1. Employee options. Consider rewarding the management board of a corpo-ration according to the performance of the corporations stock price. Financialinstruments used in this context are called employee stock options and typically arebased on European call options, where at a certain maturity, say after 5 years, theowner of the option has the right but not the obligation to purchase the stockvalue for the strike price that was fixed in advance. Thus, he will be rewarded ifthe stock exceeds the strike at a certain time in the future. Shareholders thoughtypically are interested in the performance of the stock during the hole periodand not only at fixed time points. They wish to support management decisionsthat push the stock price constantly to a high level and even more, they wish to


reward according to the level. Moreover, it is arguably fairer to reward the manage-ment board according to the performance of the stock value relative to the marketevolution. Therefore we introduce additional reference assets. Denote by S thed-dimensional stochastic process, modelling the stock of the company and d 1reference assets. Let G : Rd R be a payoff profile and let : [0, T ] Rd Rbe a reward rate function. Moreover, we incorporate a continuously paid salary bythe salary function f : [0, T ]Rd R. We suggest a class of employee options forflexible rewarding of the management board of the following type: At maturity Tthe employee obtains the payoff

G(ST ) e T0

h(Sh) dh, (20)

additionally at each instant t [0, T ], the salary

f(t, St) e

t0h(Sh) dh dt (21)

is paid. Thus, the payoff profile G may depend on the level of the stock and thereference assets, the rewarding rate and the salary function may additionally betime-dependent.

We use the following notation. For x = (x1, . . . , xd) Rd let ex := (ex1

, . . . , exd

)

and G(x) := G(ex), (, x) := (, ex) and f(, x) := f(T , ex).Let the interest rate (rt)t0 be deterministic, measurable and bounded and let

S = (S10 eL1 , . . . , Sd0 e

Ld) with a time-inhomogeneous Levy process L with localcharacteristics (b, c, F ) such that the so-called drift condition is satisfied,

bjt = rt 1

2cjjt

(exj 1 hj(x))Ft(dx), (22)

where hj is the j-th component of the truncation function h. Then, under as-sumption (A1) for some Rd with j > 1 for each component j = 1, . . . , d, the

discounted asset price process S := S e

0rs ds is a martingale, see e.g. the proof

of Proposition 4.4. in Eberlein, Jacob and Raible (2005), and S, r is a no-arbitrageasset price model driven by a time-inhomogeneous Levy process.

The following assertion shows that the model price of the employee option (20),(21) can be computed solving the related Kolmogorov PIDE.

Corollary 2.2. Let Rd such that G L2(Rd) and let the time-inhomogeneous

Levy process L satisfy assumptions (A1)(A4). Assume additionally (22) and that(A1) is satisfied for some Rd with j > 1 for each component j = 1, . . . , d.Then the time-t-value of the employee option with payout profile (20), (21),

u(T, x) := Ex

(G(LT ) e

T0(h(Lh)rh) dh +

T

0

f(T s, Ls) e

s0(h(Lh)rh) dh ds

),

is given by the unique weak solution u W 1(0, T ;H

/2 (Rd), L2(R

d))of

u+A Ttu+ Ttu = f , u(0) = G . (23)

Proof. The assertion of the proposition is an immediate consequence of Theorem1.5.

2.2. Levy-driven short rate models. As another application in finance, we con-sider bond prices in a Levy driven short rate model. In Eberlein and Raible (1999)Levy driven term structure models have been introduced first. Consider a shortrate of the form

rt := r(t, Lt) (24)

10 K. GLAU

with an Rd-valued time-inhomogeneous Levy process L and a measurable interest

rate function r : [0, T ]Rd R such that the discount factor (e

t0rh dh)0tT is

a martingale. Basic interest rate derivatives are the so-called zero coupon bonds. Atmaturity, the holder of the bond receives 1 unity of currency. In accordance withthe no-arbitrage principle, the time-t value of the zero-coupon bond with maturity0 t < T is modelled by

P (t, T ) := E(e

Tt

rh dhFt

). (25)

We reuse the notation Corollary 2.6 and with analogous arguments as for its proofwe obtain

Corollary 2.3. If L satisfies the assumptions of Corollary 2.6 with , , andj , Oj for j = 1, . . . , 2d as in Corollary 2.6, and the interest rate function r from(24) is bounded, then for every 0 t < T , the price of the zero coupon bond is givenas

P (t, T ) =

2d

j=1

uj(T t, Lt) a.s.

where uj is the unique weak solution uj W 1(0, T ;H

/2j (R

d), L2j (Rd))of

uj +A Ttuj + ruj = 0, u(0) = 1Oj . (26)

Let us furthermore consider an option on a zero-coupon bond with payoff givenby G(P (T1, T )) at the options maturity T1 (0, T ) for some measurable functionG. Since the bond prices are bounded 0 P (t, T ) P for every 0 t T , if theinterest rate function is bounded from below, it is enough to consider bounded payofffunctions G : [0, P ] [G,G]. Then, according to Proposition 2.3, G(P (T1, T2)) =G u(T1, LT1) and by the no-arbitrage principle, the time-t price of the option is

t = E(G(P (T1, T )

)e

T1t r(h,Lh) dh

Ft). (27)

Noting that uj in Corollary 2.6 is bounded for each j = 1, . . . , 2d, we obtain, againanalogue to the proof of Corollary 2.6, the following

Corollary 2.4. Under the assumptions of Proposition 2.3,

t =

2d

j=1

vj(T1 t, Lt) a.s.,

where for each j = 1, . . . , 2d, the function vj is the unique weak solution vj

W 1(0, T ;H

/2j (R

d), L2j (Rd))of

vj +A T1tvj + rvj = 0, vj(0) = u(T1, )1Oj , (28)

where u is given in Proposition 2.3.

As further applications we mention bankruptcy probabilities in the model ofAlbrecher et al. (2011), the value of barrier strategies in the bankruptcy model ofAlbrecher and Lautscham (2013) as well as reduced form modelling of credit risk,see Jeanblanc and Le Cam (2007).

2.3. Penalization of the domain. Feynman-Kac representation (19) is essentialfor the method of penalization of domain in order to derive a Feynman-Kac cor-respondence for boundary value problems. Under the presence of a dominatingdiffusion part, the method is outlined in Bensoussan and Lions (1982). In a forth-coming article, Glau (2015), the method is used for PIIACs, compare also Glau(2010). The argument is based on the following


Corollary 2.5. Let L be an Rd-valued time-inhomogeneous Levy process satis-fying assumptions (A1)(A5) for some (0, 2] and some Rd. Let f L2

(0, T ;H l(R

d))for some l 0 with l > (d )/2, let : [0, T ] Rd R be

measurable and bounded, and g L2(Rd). For each > 0 and D Rd open, the

unique weak solution u W 1(0, T ;H

/2 (Rd), L2(R

d))of

tu +A Ttu

+ Ttu + 1Dcu

= f, u(0) = g , (29)

has for every t (0, T ] almost surely the stochastic representation


Tt

h(Lh) dh e

Tt

1Dc (Lh) dh

+

T

t

f(T s, Ls) e

sth(Lh) dh e

st1Dc (Lh) dh ds

Ft) (30)

Proof. The assertion follows directly from Theorem 1.5.

2.4. Laplace transform of occupation times of Levy processes. Setting 0, f 0 and initial condition g 1, f 0, and inserting L0 = x, Corollary 2.5equation (30) becomes

u(T, x) = Ex(e

T0

1D(Lh) dh), (31)

which is the Laplace transform at of the occupation time T01D(Lh) dh the pro-

cess L spent in the domain D until time T . In Landriault, Renaud and Zhou (2011),the Laplace transforms of occupation times of spectrally negative Levy processesare discussed based on fluctuation identities. In the next corollary we show that,for a wide class of time-inhomogeneous Levy process, these transforms are char-acterized by parabolic PIDEs. Here we use the given right-hand-side and killingrate to further generalize the assumptions using Remark 2.1: We split the initial

conditions in the following way, 1 g(x) =2d

j=1 1Oj (x) a.e. with the distinct

orthants Oj of Rd. More precisely, for j = 1, . . . , 2d let pj := (pj1, . . . , pjd) with

pji {1, 1} for the 2d different possible configurations and let

Oj :={(x1, . . . , xd) R

d pjixi 0 for all i = 1, . . . , d

}.

For each j = 1, . . . , 2d we choose j Rd such that 1Oj e

j , L2(Rd). If thedistribution PLT of LT has a Lebesgue density we may rewrite equation (31) as

u(T, x) =

2d

j=1

uj(T, x), with uj(T, x) := Ex(1Oj (LT ) e

T0

1D(Lh) dh). (32)

Corollary 2.6. Let L be a time-inhomogeneous Levy process with local character-

istics (b, , F ) and > 0 with T0

|x|>1

e|x| Ft(dx) dt < . If there exist > 0

such that the symbol A of L satisfies assumption (A4) for Rd with || < , andto assure (A2) and (A3) we assume there exist constants Ci for i = 1, 2, 3 such thatfor each Rd with || < uniformly for every t [0, T ]

At( + i) C1

(1 + ||

), (33)

(At( + i)

) C2

(1 + ||

) C3

(1 + ||

)(34)

for all Rd. Let j := d1/2pj. Then, uj from equation (32) is the unique

weak solution uj W 1(0, T ;H

/2j (R

d), L2j (Rd))of

uj +A Ttuj + 1Du

j = 0, u(0) = 1Oj . (35)

12 K. GLAU

Moreover equation (32) is true, and if > 1, then for each t [0, T ], the mapping

x 7 u(t, x) := Ex(e

T0

1D(Lh) dh)is -Holder continuous with = 12 , in

particular it is continuous.

Proof. For each j = 1, . . . , 2d, the assumptions (A1)(A4) are satisfied for andj . According to part (ii) of Remark 3.3, PLT has a Lebesgue density which showsequation (32). Hence the assertion follows from Theorem 1.5 and Corollary 1.6.

2.5. Relativistic Schrodinger equation. The relation between the nonrelativis-tic Schrodinger operator and the Brownian motion is usually referred to under thenames Feynman and Kac. Carmona, Masters and Simon (1990) present withoutproof an analogous link between relativistic Schrodinger operators and Normal In-verse Gaussian Levy (NIG) processes. Baeumer, Meerschaert and Naber (2010)use this relation to model the relativistic diffusion of a particle as NIG process.We briefly present their derivation of the relativistic Schrodinger equation and theconnection to NIG processes.

The nonrelativistic Schrodinger equation for a single particle in a quantum sys-tem described by the potential energy V : Rd R+ R is the following partialdifferential equation for the wave-function : Rd R+ C,

i~

t(x, t) =

(

~2

2m+ V (x, t)

)(x, t), (36)

where i is the imaginary unit, t denotes the time derivative of , 2~ is Plancksconstant,m is the particles mass, and the Laplace operator is given by (x, t) :=d

j=12x2j

(x, t).

For a free particle i.e. V 0, a formal connection to the Kolmogorov backwardequation of the Brownian motion is obtained by the analytic continuation of theSchrodinger equation (36) in time to = it. For V 6 0, this relates equation (36)to the Kolmogorov backward equation of the Brownian motion killed with rate V .

Let us pass to the relativistic Schrodinger equation. According to Baeumer,Meerschaert and Naber (2010), the relativistic kinetic energy of a particle with restmass m and momentum p is given by

E(p) =p2c2 +m2c4 mc2 (37)

where c denotes the speed of light. The relativistic energy (37) serves as pseudodifferential operator to define the relativistic Schrodinger operator

H0()(, t) := F1(EF((, t))) (38)

for the free particle. Thus, the relativistic Schrodinger equation for a single particlein a quantum system described by the potential energy V is given by

i~

t(x, t) =

(H0 + V (x, t)

)(x, t). (39)

Analogous to the nonrelativistic case, formally inserting = it in equation (36)and setting V (x, it) := V (x, t) for every x and t, yields

t(x, t) +

(H0 + V (x, t)

)(x, t) = 0. (40)

Note that E(p) is the symbol of the NIG process L with parameters = mc2, = 0, = 1, = 0 and = c2 Idd where we use the notation of Example 3.8 andIdd denotes the identity matrix in R

d Rd.The following proposition formally justifies the Feynman-Kac for equation (40)

in terms of weak solutions.


Corollary 2.7. Let the potential energy V be measurable and bounded. Let g L2(R

d) for some Rd such that 2 m2c2. Then the unique weak solution

u W 1(0, T ;H

1/2 (Rd), L2(R

d))of

u+H0u+ V u =0, u(0) = g , (41)

has for every t (0, T ] the stochastic representation


Tt

VTh(Lh) dhFt

)a.s. (42)

Corollary 2.7 is a direct consequence of Theorem 1.5 and Example 3.8.

3. Examples of classes of time-inhomogeneous Levy processes

First, we provide some general assertions concerning the conditions of Feynman-Kac Theorem 1.5. Then we consider Levy processes with a Brownian componentfollowed by pure jump Levy processes. We conclude section presenting two con-struction principles that yield a rich class of time-inhomogeneous Levy processessatisfying the assumptions of Theorem 1.5. We introduce a short notation for theassumptions and denote

(A1) (EM(R),(A2) (Cont-A) for and every R,(A3) (Gard-A) for and every R,(A4) t 7 At( i) for every R

d is cadlag for each R.

Remark 3.1. Let A is the symbol of a time-inhomogeneous Levy processes sat-isfying (EM(R). By virtue of Lemma A.1 and the continuity of Levy symbolsvalidity of (A2) for A is equivalent to the following asymptotic condition: For everyN > 0 there exist a constant G > 0 such that

(At( + i)

) G|| A(i) for every Rd such that || > N.

In Eberlein and Glau (2014), examples of PIIACs satisfying assumptions (A1)(A3) are discussed. For = 0, assumptions (A2) and (A3) are studied in Glau(2014) for the case of Levy processes. The symbol of a Levy process is constant intime, hence assumption (A4) is trivially satisfied for their symbols.

We start with some general assertions that can be read as construction principlesfor Levy processes with symbols satisfying (A1)(A4).

Remark 3.2. For j = 1, 2 let Lj be two stochastically independent time-inhomo-geneous Levy processes with symbol Aj such that (A1)(A4) are satisfied for somej and the same Rd. Then the sum L := L1 + L2 is a (time-inhomogeneous)Levy process with symbol A := A1 + A2, and (A1)(A4) are satisfied for :=max(1, 2). Compare Remark 4.5. in Glau (2014) for the case = 0 and condi-tions (A1)(A3).

It is obvious that conditions (A2) and (A3) are not satisfied for every time-inhomo-geneous Levy process and not for every Levy process. On the one hand, the natureof the class of processes satisfying such a continuity and Garding condition can becharacterized by its distributional properties:

Remark 3.3. Let L be a time-inhomogeneous Levy process with symbol A =(At)t0 and let R

d. If condition (Gard-A) is satisfied for some (0, 2],we have for 0 s t T ,

e

tsAu(i) du

C1 e(ts)C2||

. (43)

In particular, (Gard-A) implies for every t (0, T ] that the distribution of Lt hasa smooth Lebesgue density.

14 K. GLAU

On the other hand, continuity and Garding condition (A2) and (A3) relate to thepath behaviour of the process:

Remark 3.4. A Levy process with symbol A satisfying (A2) and (A3) for some (0, 2) has Blumenthal-Getoor index , see Glau (2014). Hence, every purejump Levy process satisfying assumptions (A2) and (A3) has infinite jump activity.In particular, compound Poisson processes do not satisfy (A3). Variance Gammaprocesses have Blumenthal-Getoor index 0 and thus also do not satisfy both of theconditions, (A2) and (A3), compare Example 4.4 in Glau (2014).

For the Variance Gamma process, the small jumps may be approximated by ascaled Brownian motion as introduced in Asmussen and Rosinski (2001). Thus, thepure jump process is approximated by a series of jump diffusion Levy processes, aclass for which assumptions (A2) and (A3) are satisfied:

Example 3.5. [Multivariate Levy processes with Brownian part] Rd-valued Levyprocesses L with characteristics (b, , F ) with a positive definite matrix and aLevy measure satisfying (A1). Then the symbol of the process satisfies (A2) and(A3) with = 2, compare Example 4.6 in Glau (2014). For the time-inhomogeneousextension, see section 3.2.

3.1. Pure jump Levy processes and operators of fractional order. Typ-ically, either the Levy measure or the characteristic function of a Levy processis known explicitly. Our first example provides sufficient conditions on the Levydensity under which the main assumptions are satisfied.

Example 3.6. [Levy processes with Levy density] Let L be a real-valued Levyprocess and a special semimartingale with characteristic triplet (b, 0, F ) with respectto the truncation function h(x) = x and fix some R. Let the Levy measureF with

|x|>1

e|x| F (dx) < have a Lebesgue density F (dx) = f(x) dx. For its

symmetric part fs(x) := 1/2(f(x) + f(x)) assumefs(x) =

C|x|1+Y + g(x) with g(x) = O

(|x|1Y +

)for x 0

with 0 < . In the following cases, the symbol of L satisfies (A2) and (A3) withSobolev index = Y and weight .

a) Let 1 < Y < 2.b) The antisymmetric part fas(x) := f(x) fs(x) of the Levy density f

satisfies fas(x) = O(|x|1

)for x 0

1) with < Y = 1, or2) let 0 < Y < 1,

|x|f(x) dx < , and moreover b =

xF (dx).

For = 0 the assertion is proved in Proposition 4.14 in Glau (2014). The case 6= 0 thus follows by Lemma 3.2.

Example 3.7. [Univariate generalized tempered stable Levy process] A generalizedtempered stable Levy process L with parametersC, C+ 0 such that C+C+ > 0and G, M > 0 and Y, Y+ < 2, is a pure jump Levy process L whose Levy measureF temp is given by its Lebesgue density

f temp(x) =

{ C|x|1+Y

eGx for x < 0C+

|x|1+Y+eMx for x 0 ,

compare Poirot and Tankov (2006). For C = 0 we set Y := 0 and denote by(b, 0, F ) the characteristic triplet of L with respect to some truncation function h.Example 4.15 in Glau (2014) shows that in each of the following cases, assumptions(A1)(A3) are satisfied for := max{Y+, Y} and (G,M).

(i) = max{Y+, Y} > 1.


(ii) Y := Y = Y+ = 1 and C = C+.(iii) 0 < = max{Y+, Y} < 1 and b := C+M

1Y+(Y+) CG1Y(Y)

for the truncation function h(x) = x.

In the following example, validity of the assumptions is derived from the closedform of the characteristic function of the Levy process.

Example 3.8. [Multivariate Normal Inverse Gaussian (NIG) processes] Let Lbe an Rd-valued NIG-process i.e. a Levy process with L1 = (L

11, . . . , L

d1)

NIGd(, , , ,) with parameters , 0, , Rd and symmetric positive

definite matrix Rdd with 2 > ,. Then the symbol of L is given by

A(u) = iu, (

2 , 2 + iu,( iu)

),

where by , we denote the product z, z =d

j=1 zjzj for z C

d, compare e.g.

equation (2.3) in Hammerstein (2010).Assumptions (A1*), (A2) and (A3) are satisfied for the index = 1 for any Rd such that 2 > ,( ) for all R, see Example 7.3 in

Eberlein and Glau (2014). This is in particular the case, if 2 + 2 2

.

Further examples are discussed in Examples 4.84.10 in Glau (2014): For = 0,generalized student-t processes and Cauchy processes satisfy assumptions (A2) and(A3) with = 1. And for multivariate -semi stable with (1, 2] and univariatestrictly stable Levy processes with = 1, (A2) and (A3) are satisfied with index and = 0. Also for univariate generalized hyperbolic processes, validity ofassumptions (A2) and (A3) can be derived with index = 1, see Example 7.2 inEberlein and Glau (2014).

3.2. Time-inhomogeneous processes. Time-inhomogeneous Levy processes arisenaturally in financial applications because they provide a considerably better fit tothe time-evolution of data than Levy processes. We present two construction prin-ciples and conditions under which the resulting time-inhomogeneous ones satisfy(A1)(A4). A natural family of time-inhomogeneous Levy processes is obtainedby inserting time-dependent parameters into a given parametric class of Levy pro-cesses. It is straightforward to show the following

Lemma 3.9. Let (A(p, ))pP a parametrized family of symbols. Fix some Rd

and some (0, 2). Let (A2) and (A3) be satisfied for A, uniformly for all p P.Then for t 7 p(t) measurable, (A2) and (A3) are satisfied for

At() := A(p(t), ) for t [0, T ] and U.

If moreover, (p, ) 7 A(p, ) and t 7 p(t) are continuous, (At)t0 is the symbol ofa time-inhomogeneous Levy process L and satisfies also (A4). If additionally (A1)is satisfied for L, then it is also satisfied for L.

As another natural construction we consider stochastic integrals of deterministicfunctions with respect to Levy processes. Let L be an Rd-valued Levy process.Then

Xt := f Lt :=

t

0

f(s) dLs :=

(d

k=1

t

0

f jk(s) dLks

)

jd

with a deterministic and L-integrableRnd-valued function f is an Rn-valued semi-martingale with deterministic characteristics. Denote by (b, c, F ) the characteristicsof L w.r.t. the truncation function h. Then, by elementary arguments based on

16 K. GLAU

their definition, the characteristics (bXt , cXt , F

Xt )t0 of X w.r.t. h are shown to be

given by

bXt = f(t)b+

Rd

(h(f(t)x) f(t)h(x)

)F (dx),

cXt = f(t)cf(t)tr, (44)

FXt (B) =

Rd

1B

(f(t)x

)F (dx) for every B B

(R

d \ {0}).

In particular, X is a time-inhomogeneous Levy process in the sense of our definitionif integrability condition (6) is satisfied for its characteristics. Moreover, if A denotesthe symbol of L, the symbol AX of X is given by

AXt () = A(f(t)tr

)+ i, b(h, h, f) for every Rd. (45)

where b(h, h, f) :=Rd

(h(f(t)x) f(t)h(x)

)F (dx). This generalizes Example 7.6

in Eberlein and Glau (2014) for the case f : [0,) R+.

Lemma 3.10. Let L a Levy process and a special semimartingale with symbol A.Let f : [0,) Rnd measurable and such that there exist constants 0 < f, f

with

sup0tT

f(t)f(t)tr1/2 f and sup0tT

f(t)f(t)tr1/2 f, (46)

where denotes the spectral norm. Then X := f L is a time-inhomogeneousLevy process and a special semimartingale with symbol

AXt () = A(f(t)tr

)for all Rn.

Fix some > 0, X Rd with |X | f and some > 0. If E e|Lt| < for

some, respectively for all t > 0, then X satisfies (EM(RX )). If additionally A

satisfies (A2) and (A3) for every weight Rd with || and index > 0, then(A2) and (A3) hold for AX with the same index and weight X . Moreover, if(A4) holds for A it also satisfied for AX .

Proof. From the assumptions it is immediate that f is integrable with respect toL and hence X is a semimartingale with characteristics of form (44). Integrabilitycondition (6) also follows directly, so X is a time-inhomogeneous Levy process.Since L is a special semimartingale, where

|x|>1

|x|F (dx) < where F denotes

the Levy measure of L and (46) implies

T

0

|x|>1

|x|Ft(dx) Tf

|x|>1/f

|x|F (dx) < (47)

which shows that also X is a special semimartingale. Therefore we may choose hand h as the identity so that b(h, h, f) = 0 and from (45) we obtain the equalityAXt () = A

(f(t)tr

). The assertion on the exponential moment condition (A1)

follows analogously to (47), and the assertions on (A2)(A4) are immediate conse-quences of (46), the continuity Levy symbols and Lemma A.1.

Combining the examples of Levy processes with Lemma 3.10, one obtains variousclasses of PIIAC satisfying our main assumptions. Here we consider the following


Example 3.11. [Time-inhomogeneous Levy jump diffusion] Let L a PIIAC whosepure jump parts symbol and Levy measure satisfy (A1), (A2), (A4) for some Rd

and some [0, 2) as well as assumption (A3) for and = 2 and has a time-dependent covariance matrix (t) such that the mapping t 7 (t) from [0, T ] tothe positive definite d d-matrices is continuous, then the symbol of the processsatisfies (A1)(A4) for weight Rd and Sobolev index = 2.

4. Robustness of the weak solutions

We provide the following general robustness result, a special case of which iscrucial for our derivation of the Feynman-Kac Theorem 1.5.

Let X H X and Y H Y be two Gelfand triplets such thatY X . For t [0, T ] and each n N let A nt resp. A t be an operator withassociated real-valued bilinear form ant resp. at. We introduce the following set ofassumptions.

(An1) There exists constants Ci > 0 for i = 1, 2, 3 such that uniformly for alln N and all t [0, T ],

at(u, v) C1uXvX for all u, v X, (48)

min{ant (u, u), at(u, u)} C1u2X C2u

2H for all u X. (49)

(An2) For each n N there exists a constant Ci(n) > 0 for i = 3, 4, 5 such thatuniformly for all t [0, T ],

ant (u, v) C3(n)uY vY for all u, v Y, (50)

an(u, u) C4(n)u2Y C5(n)u

2H for all u Y. (51)

(An3) There exists a sequence C6(n) 0 for n such that uniformly for allt [0, T ],

(ant at)(u, v) C6(n)uXvX for all u, v X. (52)

Proposition 4.1. Let the operators A and A n for n N satisfy (An1)(An3).Let fn, f L2(0, T ;H) with fn f in L2

(0, T ;H) and gn, g H with gn g in

H. Then the sequence of unique weak solutions un W 1(0, T ;Y,H) of

un +A nt un = fn, un(0) = gn (53)

converges strongly in L2(0, T ;X) C(0, T ;H). Moreover, the limit u belongs to

W 1(0, T ;X,H) and is the unique weak solution of

u+A tu = f, u(0) = g. (54)

Proof. We insert the weak solution un of equation (53) as test function in (53). Us-

ing t0

(un(s), u(s)

)ds = 12

(un(t)2Hu

n(0)2H), compare Wloka (1987), equation

(2) on p. 394, inserting Garding inequality (49) and the inequality of Young, in-tegrating over time and applying the lemma of Gronwall yields the existence ofconstants C1, C2 > 0 with

supt[0,T ]

un(t)2H + C1un2L2(0,T ;X) C2

(f2L2(0,T ;X) + g

2H

). (55)

18 K. GLAU

Subtracting equation (53) for n and m and inserting wnm := un um with thesolutions un and um of equation (53) for n and m as test function yields

T

0

[(wnm(t), wnm(t)) + ant (w

nm(t), wnm(t))]dt

=

T

0

[fn(t) fm(t), wnm(t)H (a

nt a

mt )(u

m(t), wnm(t))]dt.

(56)

Thanks to inequality (52) and (55) and Youngs inequality we obtain

T

0

(ant amt )(um(t), wnm(t)) dt nm + wnm2L2(0,T ;X)

with 0 < , 0 < nm 0 for n,m . Inserting the last inequality in equation(56), applying again Wloka (1987), equation (2) on p. 394, Garding inequality (49),the inequality of Young and the lemma of Gronwall yield the existence of constantsC3, C4 > 0 with

supt[0,T ]

wnm(t)2H + C3wnm2L2(0,T ;X)

C4(nm + fn fm2L2(0,T ;X) + g

n gm2H),

from where the strong convergence of the sequence un in the space L2(0, T ;X)

L(0, T ;H) follows. We have

T

0

u(t), v(t)H dt+

T

0

at(u(t), v(t)

)dt

=

T

0

un(t), v(t)H dt+

T

0

ant(un(t), v(t)

)dtRn(un u, v)

with

Rn(, v) :=

T

0

(t), v(t)H dt+

T

0

(at ant )((t), v(t)

)dt

+

T

0

at((t), v(t)

)dt.

Due to the convergence un u in L(0, T ;H) L2

(0, T ;X), from equation (52)

and from the continuity of a we get Rn(unu, v) 0 for n and hence insertingequation (53), we have

T

0

u(t), v(t)H dt+

T

0

at(u(t), v(t)

)dt

=

T

0

(f(t), v(t)

)dt+ g, v(0)H

(57)

for all v C1([0, T ];Y

)with v(T ) = 0. From the density of Y in X and from

Proposition 23.20 in Zeidler (1990) we deduce that u W 1(0, T ;X,H) and u


solves (54). Moreover, due to inequality (48) and (49), equation (54) has a uniqueweak solution in W 1(0, T ;X,H) and the assertion of the theorem follows.

5. Proof of the Feynman-Kac Theorem 1.5

The key steps in the proof of Feynman-Kac Theorem 1.5 are first the applicationof Itos formula with the help of part (ii) of the following regularity assertion, secondthe convergence of the regularized solution due to robustness result Proposition 4.1,and third Lemma 5.2 linking convergence in L2(R

d) respectively L2(0, T ;H l(R

d))

to convergence of conditional expectations.

Lemma 5.1. Fix some Rd and some > 0. Let A a pseudo differentialoperator whose symbol A has for each t [0, T ] a continuous extension on U

that is analytic in the interior

U and satisfies inequality (16) and has Sobolevindex and the mapping t 7 At( i) is continuous of every R

d. Let

for some L([0, T ] Rd), g L2(Rd), f L2

(0, T ;H

/2 (Rd)

)and u

W 1(0, T ;H

/2 (Rd), L2(R

d))the unique weak solution of

u+A Ttu+ Ttu = f, (58)

u(0) = g. (59)

Then the following assertions are valid.

(i) Let m 1. If g H(m1)/2 (Rd), f L2

(0, T ;H

(m1)/2 (Rd)

)and for every

1 k m we have h L2(0, T ;H

k/2 (Rd)

)for all h L2

(0, T ;H

k/2 (Rd)

),

then u L2(0, T ;H

m/2 (Rd)

)and u L2

(0, T ;H

(m2)/2 (Rd)

).

(ii) If g H (Rd) for = m+ d/2 + max(, 1/2), f L2

(0, T ;H (R

d))for

= m + (d + 1)/2 and C0 ([0, T ] Rd), then for every multiindex

k = (k1, . . . , kd) with |k| m the derivative (1+t)Dku is in C([0, T ]Rd).

If moreover A is the infinitesimal generator of a Levy process and f iscontinuous, then equality (58) holds pointwise for every (t, x) (0, T ]Rd.

Proof. We derive the regularity assertion by explicit operations on the Fourier

transform of the solution of (58)(59). Let u W 1(0, T ;H

/2 (Rd), L2(R

d))be

the unique weak solution of (58)(59) and u = u1 + u2 + u3, with

F(u1(t)

):= F(g) e

TTt

Au(i) du,

F(u2(t)

):=

t

0

F(f(s)

)e

TsTt

Au(i) du ds,

F(u3(t)

):=

t

0

F(u(s)

)e

TsTt

A(i) d ds

and hence

tF(u1(t)

)= ATt( i)F

(u1(t)

),

tF(u2(t)

)= ATt( i)F

(u2(t)

)+ F

(f(t)

), (60)

tF(u3(t)

)= ATt( i)F

(u3(t)

)F

(u(t)

).

In particular, u satisfies (58). From inequality (43) with constants C1, C2 > 0 andthe inequality of Cauchy-Schwarz follows the existence of a constant c1, c2 > 0 s.t.

20 K. GLAU

for every (t, ) [0, T ]Rd,F

(u1(t)

)()

C1F(g)()

etC2||

F(uj(t)

)()

C1

t

0

F(f(s))()2 ds

1/2

t

0

e(ts)2C2||

ds

1/2

c2

T

0

F(f j(s))()2(1 + ||

)ds

1/2

,

as well asF

(tu

1(t))()

c1F(g)()

(1 + ||)

etC2||

,

F(tu

j(t))()

c2

T

0

F(f j(s))()2(1 + ||

)ds

1/2

+F(f j(s))()

,

for j = 1, 2 with f1 = f and f2 = u. Hence there exist constants c4, c5 > 0 with

uL2(0,T ;H

m/2 (Rd))

c4(g

H(m1)/2 (Rd))

+ fL2(0,T ;H

(m1)/2 (Rd))

+ uL2(0,T ;H

(m1)/2 (Rd))

)

as well as

tuL2(0,T ;H(m2)/2 (Rd))

c4(g

H(m1)/2 (Rd))

+ fL2(0,T ;H

(m1)/2 (Rd))

+ uL2(0,T ;H

(m1)/2 (Rd))

).

For m = 1, inserting u L2(0, T ;H

/2 (Rd)

)as well as u L2

(0, T ;H

/2 (Rd)

)

we obtain u L2(0, T ;H

/2 (Rd)

)and tu L

2(0, T ;H

/2 (Rd)

). In particular,

u W 1(0, T ;H

/2 (Rd), L2(R

d))is the unique weak solution u = u of equation

(58)(59).

For m = 2 it is thus sufficient to notice that u L2(0, T ;H

/2 (Rd)

)yields

u L2(0, T ;H

/2 (Rd)

)as well as tu L

2(0, T ;L2(R

d)). Iteration shows part (i)

of the Lemma.(ii) By the inequality of Cauchy-Schwarz and

Rd

(1+||

)dd < for > 0,

we obtain for = m+ d/2+max(, 1/2) and = m+ (d+1)/2 the existence of aconstant c3 > 0 s.t.

Rd

(1 + t)F

((u1 + u2)(t)

)()

(1 + ||)m

d

c3(gH (Rd) + fL2(0,T ;H

(Rd))

)< .

Moreover, the mappings t 7 F(u(t)

)() and t 7 tF

(u(t)

)() are continuous for

each Rd. Dominated convergence yields Dkx(1 + t)(u1 + u2) C([0, T ]Rd)

for every multiindex k = (k1, . . . , kd) with |k| 0.Moreover, there exists a constant c4 > 0 such that

Rd

(1 + t)F

(u3(t)

)()

(1 + ||)m

d c4uL2(0,T ;H (Rd)) <

and dominated convergence yieldsDkx(1+t)u3 C([0, T ]Rd) for every multiindex

k = (k1, . . . , kd) with |k| 0.In order to derive (58) pointwise, fix a t T for which the equation holds as

operator equation over L2 and choose a sequence un C0 ((0, T )R

d) such that


un(t) u(t) in the norm of H/2 (Rd) and un(t) u in the norm of L

2(R

d).

Moreover, let C0 (Rd). Since u(t) C2(Rd), Au is defined pointwise and an

elementary manipulation shows

Rd

Au(x)(x) e2,x dx = u,A,L2 = limnun,A

,L2

with the adjoint operator A, defined in Lemma A.2. By equation 7, LemmaA.2, continuity of the scalar product and of the bilinear form, we obtain

limn

un,A,L2 = limn

a(un, ) = a(u, )

and hence

u, L2 + A u, L2 = f, L2 for all C0 (R

d).

Thanks to the fundamental lemma of variational calculus the equality follows for tand a.e. x Rd. Since we can choose t arbitrary from a dense subset in (0, T ) theassertion follows by continuity of u+A u f .

Lemma 5.2. Let L be a PIIAC with symbol A = (At)t[0,T ] such that satisfies

EM(R) for some Rd and (Gard-A) for some > 0 and every R.

Then

(i) for t > 0 there exists a constant C(t) > 0 with E[|(Ls)|] C(t)L2(Rd)

uniformly for all L2(Rd) and s [t, T ],

(ii) for l > (d )/2 and every 0 t < T there exists a constant C1 > 0

withE

( Tt (s, Ls) ds

Ft) C1L2(t,T ;Hl(Rd)) uniformly for all

L2(0, T ;H l(R

d)).

Proof. (i) By Remark 3.3, assumption (A3) yields that the distribution of Lt hasa Lebesgue density. Applying Parsevals identity, see e.g. equality (10) on p. 187in Rudin (1966), we obtain

E|(Lt)| =1

(2)d

Rd

F(||)( i)e t0As(i) ds d,

inserting inequality (43) and the inequality of Cauchy-Schwarz yields assertion (i).

(ii) W.l.o.g. 0. We have E( T

t (s, Ls) dsFt

)= G(Lt) with

G(y) = E( Tt

0

(s+ t, Lt+s Lt + y) ds).

The theorem of Fubini and Parsevals identity yield

G(y) =1

(2)d

Rd

Tt

0

F(y(s+ t)

)( i)e

s0At+u(i) du ds d

where yf(x) := f(x + y). Notice that F(yf)() = e,yF() inserting the

inequality of Cauchy-Schwarz and equation (43) with constants C1, C2 > 0, we

22 K. GLAU

obtain for l > d some constants c1, c2 > 0 with

|G(y)| C1

Rd

Tt

0

F(y(s+ t)

)( i)

2 dsTt

0

e2sC2||

ds

1/2

d

c1

T

0

Rd

F((s+ t)

)( i)

(1 + ||)

d ds

c2L2(0,T ;Hl/2 (Rd)).

We are now in a position to prove part (ii) of Theorem 1.5.

Proof. First, let t 7 At( i) continuous for every Rd. By density ar-

guments, we choose sequences gn C0 (Rd), fn C0

([0, T ] Rd

)as well as

n C0([0, T ]Rd

)s.t. each partial derivative of is bounded and such that for

n

gn g in L2(Rd),

fn f in L2(0, T ;H l(R

d)),

n in L([0, T ]Rd

).

Let un W 1(0, T ;H

/2 (Rd), L2(R

d))be the unique weak solution of

un +A tun + nt u

n = fn, un(0) = gn. (61)

Proposition 4.1 yields the convergence un u in L2(0, T ;H

/2 (Rd)

) C

(0, T ;L2(R

d))

to the weak solution u W 1(0, T ;H

/2 (Rd), L2(R

d))of

u+A tu+ tu = f, u(0) = g. (62)

Lemma 5.1 shows that the equality holds pointwise and that u is regular enoughto apply Itos formula. Therefore let wn(t, x) := un(T t, x) and

(bt, t, Ft

)t[0,T ]

denote the local characteristics of L, then Itos formula for semimartingales, seee.g. Theorem I.4.57 in Jacod and Shiryaev (2003) yields

wn(T, LT ) e

T0

n(L) d wn(s, Ls) e

s0n(L) d

=

T

s

[wn A hw

n wn](h, Lh) e

h0

n(L) d dh

+

T

s

(1/2h w

n(h, Lh))e

h0

n(L) d dWh

+(e

0n(L) d

(wn(, L + x) w

n(, L))1(s,)()

)(

)T.

Thanks to our assumptions on gn, fn and n, we may decompose un in threesummands along the lines of (60) and application of part (ii) of Lemma 5.1 it iselementary to conclude that wn and wn belong to L2(Rd). Hence, the integralswith respect to W and are martingales, compare Theorem II.1.33 a) in

Jacod and Shiryaev (2003). Inserting the identity wn A hwn nwn = f

nwith

f(t, x) := fn(T t, x), multiplication of the equation with the term e

s0n(L) d


and taking the conditional expectation yields for 0 s T ,

E(wn(T, LT ) e

Ts

n(L) dFs

) wn(s, Ls)

= E( T

s

fn(h, Lh) e

hs

n(L) d dhFs

).

(63)

Let w.l.g. 0 < s T . We derive the stochastic representation by letting n foreach term in (63):

Denote w(t, x) := u(T t, x). From the convergence wn(s, ) w(s, ) in L2(Rd)

and Lemma 5.2 (i) for s > 0, we get the convergence

wn(s, Ls) w(s, Ls) in L1(P ) and a.s for a subsequence.

Since n converges to in L, dominated convergence yields ban(L) d b

a(L) d and uniform boundedness of the sequence for 0 a b T .

Together with wn(s, Ls) w(s, Ls) in L1(P ) the convergence

E(wn(t, Lt) e

Ts

n(L) d w(t, Lt) e

Ts

(L) dFs

) 0

for n follows elementary using the triangle inequality.Next, denote f(t, x) := f(T t, x). From part (ii) of Lemma 5.2, there exists a

constant c2 > 0 for l > (d )/2 with

E( T

s

(fn f)(h, Lh) dh

Fs) c1fL2(t,T ;Hl(Rd)) 0

due to fn f L2(t, T ;H l(R

d)). Again from elementary application of the

triangle inequality we obtain the convergence of the second line in equation (63)and thus the assertion of Theorem 1.5 under the additional assumption that themapping t 7 At( i) is continuous for every R

d. Thanks to the tower ruleof conditional expectations and that existence and uniqueness of the weak solutiondoes not require continuity of the bilinear form, the claim follows by induction overthe continuity periods also under the more general assumption that t 7 At( i)is cadlag for every Rd.

6. Acknowledgement

The roots of the present paper go back to the authors dissertation Glau (2010)and the author thanks Ernst Eberlein for his valuable support and the DFG forfinancial support through project EB66/11-1. For fruitful discussions on furtherdevelopments the author expresses her gratitude to Carsten Eilks and to ClaudiaKluppelberg for rich comments on a previous version of the manuscript.

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Cont, R., Lantos, N., Pironneau, O., 2011. A reduced basis for option pricing. SIAMJournal on Financial Mathematics 2 (1), 287316.

Cont, R., Tankov, P., 2004. Financial Modelling With Jump Processes. FinancialMathematics. Chapman & Hall/CRC Press.

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Appendix A. Adjoint Operator

For a Levy process L with characteristics (b, , F ;h) we denote by A (b,,F ) andA(b,,F ) its Kolmogorov operator and its symbol.

Lemma A.1. Let Levy process with characteristics (b, , F ;h) satisfy EM() andlet A , A its Kolmogorov operator and symbol. Then

A() := A( + i) = A(b ,,F)() +A(i) for all Rd

with

b = b+ +

Rd

(1 e,y

)h(y)F (dy),

F (dy) = e,y F (dy)

i.e. A is the symbol of a Levy process with killing rate A(i). Moreover, itsKolmogorov operator A satisfies

A = e,A (e, ) = A (b ,,F)+A(i) for all C0 (R

d).

Proof. It is elementary to verify the assertion on the symbol which can be nicelyused to verify the assertion for the operator: Let C0 (R

d) then F(e, )() =

http://dblp.uni-trier.de/db/journals/rda/rda2.html#Pironneau11http://www.esaim-m2an.org/article_S0764583X12000398

26 K. GLAU

F()( i) and

A(e,

)(x) = F1

(AF(e, )

)

=1

(2)d

Rd

ei,x A()F()( i) d

=e,x

(2)d

Rd

ei,x A( + i)F()() d.

For all C0 (Rd) let

F() := e, F

( e,

)and F1 () := e

, F1( e,

).

Theorem 4.1 in Eberlein and Glau (2014) shows that for pseudo differential operatorA whose symbol A has a continuous extension to U that is analytic in the interiorof U and satisfies the continuity condition (CA),

A = F1(A) = F1 (AF()) for all C

0 (R

d) (64)

and Parsevals equality yields for all , C0 (Rd),

a(, ) = A, L2 =1

(2)dAF(),F()L2 . (65)

We denote by A , and A, the L2-adjoint of the pseudo differential operator A

and its symbol, given for all , C0 (Rd) by

A , L2 = ,A,L2 , (66)

AF(),F()L2 = F(), A,F()L2 . (67)

Lemma A.2. Let L a Levy process with characteristics (b, , F ;h) satisfies thatEM() and let A , A its Kolmogorov operator and symbol. Then

A, = A = A(b,,,F,) +A(i),

A, = e,A(e,

)= A (b

,,,F,)+A(i),

where

b, = b,

F,(B) = Fsym(B) Fasym(B), for all Borel sets B 6= {0}.

where Fsym(B) =12F

(B) + F(B) and Fasym(B) = F(B) Fsym(B).

Moreover F, is a Levy measure.

Proof. For every C0 (Rd) we have

AF(),F()L2 = AF

(e,

),F

(e,

)L2

= F(e,

), AF

(e,

)L2 ,

and by Lemma A.1 and since A(z) R for (z) Rd,

A = A(b ,,F) +A(i).

Since A(b ,,F) is the symbol of a Levy process,

A(b ,,F)() = A(b ,,F)() for all Rd,

from where the assertion of the lemma follows directly.


Appendix B. Proof of Theorem 1.3

By the assumption on the analyticity of A and (16) we obtain from Theorem 4.1in Eberlein and Glau (2014) for all t [0, T ], all R and every , C

0 (R

d),

at(, ) =1

(2)dAtF(),F()L2 =

1

(2)dAt( i

)F(),F()L2

=1

(2)d

Rd

At( i)F()( i)F()( i) d (68)

This equality directly entails that (Cont-A) implies (Cont-a). Additionally, togetherwith the following elementary inequalities it yields the implication of (Gard-a) by(Gard-A): For C1 > 0, C2 0, 0 < and 0 < C3 < C1 there exits a constantC4 > 0 such that C1x

C2x C3x

C4 for every x 0 and

C2||2 C3(1 + ||

2) C2||2 C3(1 + ||

2) c2(1 + ||)2 c3 (69)

with a strictly positive positive constant c2 and C3, c3 0.

Moreover, piecewise continuity of t 7 at(u, v) for every u, v H (R

d) followsfrom the piecewise continuity of t 7 At(z) for every z U and dominatedconvergence thanks to (Cont-A).

In order to derive implication (i) (ii), let us first note that following closely thederivation of the fundamental lemma of variational calculus yields for any continu-ous function , that if for all u H(R

d) such that F(u) is compactly supported

Rd

()|F(u)()|2 e2

, d 0 (70)

holds, then () 0 for all Rd. To this end, let us for a moment assume() < 0 for some Rd. Thanks to the continuity of the function, the integrandwould be negative on a nonempty open subset of U Rd. We may choose a functionu such that its Fourier transform F(u) is smooth, not constant and such that itscompact support is contained in U . Noting that those functions lie in H(R

d), we

would obtain a contradiction to inequality (70).Since (Cont-a) implies inequality (70) for the continuous mappings 7 C(1 +

||)2 (A( i)

)and 7 C(1+ ||)2

(A( i)

)for all t [0, T ] and all

R, (Cont-A) follows. Similarly, using once again inequality (69), we obtainthat (Gard-a) implies (Gard-A).

Finally, we observe that limst as(u, u) = at(u, u) implies

limst

Rd

As( i)F(u)()

2 d =

Rd

At( i)F(u)()

2 d

while on the other hand dominated convergence shows

limst

Rd

As( i)F(u)()

2 d =

Rd

limst

As( i)F(u)()

2 d.

Hence, (70) yields limst As( i) = At( i

) for all Rd and all R, sopiecewise continuity of the bilinear form entails piecewise continuity of the symbol.

1. Introduction2. Applications2.1. Employee options2.2. Lvy-driven short rate models2.3. Penalization of the domain2.4. Laplace transform of occupation times of Lvy processes2.5. Relativistic Schrdinger equation

3. Examples of classes of time-inhomogeneous Lvy processes3.1. Pure jump Lvy processes and operators of fractional order3.2. Time-inhomogeneous processes

4. Robustness of the weak solutions5. Proof of the Feynman-Kac Theorem 1.56. AcknowledgementReferencesAppendix A. Adjoint OperatorAppendix B. Proof of Theorem 1.3

FEYNMAN-KAC FORMULA FOR LEVY PROCESSES ´ WITH DISCONTINUOUS KILLING RATE

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