Penalization methods for the Skorokhod problem and reflecting SDEs with jumps

Penalization methods for the Skorokhod problem and reflecting SDEs with jumps

Abstract

We study the problem of approximation of solutions of the Skorokhod problem and reflecting stochastic differential equations (SDEs) with jumps by sequences of solutions of equations with penalization terms. Applications to discrete approximation of weak and strong solutions of reflecting SDEs are given. Our proofs are based on new estimates for solutions of equations with penalization terms and the theory of convergence in the Jakubowski -topology.

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0 \volume19 \issue5A 2013 \firstpage1750 \lastpage1775 \doi10.3150/12-BEJ428 \newremarkremarkRemark[section] \newremarkexample[remark]Example

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Penalization methods

{aug}

and

penalization methods \kwdreflecting stochastic differential equation \kwd-topology \kwdSkorokhod problem

1 Introduction

Let be a convex open set in . Consider a -dimensional reflecting stochastic differential equation (SDE),

(1)

where is a -dimensional semimartingale with , is an adapted process with , and is a continuous function such that

(2)

(for the precise definition, see Section 3). Our main purpose is to study the problem of approximation of weak solution of (1) by solutions of nonreflecting SDEs of the form

(3)

where and are perturbations of and , respectively, and denotes projection of on . Because for large , the drift term forces to stay close to , it is called the penalization term, and the SDE (3) is called the SDE with penalization term.

The foregoing problem was intensively investigated in the case where is a Lipschitz continuous function, , and is a continuous semimartingale. In particular, Lions et al. [16] and Menaldi [20] have proven that for , provided that is a -dimensional standard Wiener process. In the case where has jumps, to the best of our knowledge, such a problem has been considered previously only by Menaldi and Robin [21] and Łaukajtys and Słomiński [15]. Menaldi and Robin studied the case where is a diffusion with Poissonian jumps and . However, they imposed a very restrictive condition on the Poissonian measure coefficient, and consequently, is a process with continuous trajectories. In this case, earlier methods of approximation remain in force. In earlier work, we considered in detail the case where and is a general semimartingale. Because the approximating sequence might not be relatively compact in the Skorokhod topology , we proved our convergence results in the -topology introduced by Jakubowski [10]. It is worth pointing out that in both of the aforementioned papers, the initial process is constant (i.e., ), and is a Lipschitz continuous function.

The purpose of the present paper is to investigate the problem of approximation of by in the case of arbitrary initial process and arbitrary continuous coefficient satisfying the linear growth condition (2). Our proofs are based on new estimates for solutions of equations with penalization terms.

The paper is organized as follows. In Section 2 we consider a deterministic problem of approximating a solution of the Skorokhod problem , on domain associated with a given function such that (for precise definition, see Section 2). The penalization method involves approximating by solutions of equations of the form

(4)

where in the Skorokhod topology . Lions and Sznitman [17] and Cépa [5] proved that tends to if is continuous. We omit the latter assumption and consider arbitrary function . In this general case, we prove that the variation of the penalization term of the SDE (4) is locally uniformly bounded and for fixed , provided that , which implies in particular that tends toward in the -topology. It is noteworthy that, similar to [5], here we do not assume that the domain satisfies the so-called condition () introduced by Tanaka [29].

In Section 3 we present new estimates on solutions of equations with penalization terms associated with a given process such that , that is, solutions of SDEs,

(5)

In particular, we prove that if is a process admitting the decomposition , where is an adapted process, is an adapted local martingale with and is an adapted processes of bounded variation with , then, for every , there exist constants such that for every ,

and

where denotes the usual modulus of continuity and .

In Section 4, we use estimates derived in Section 3 to prove our main results on the approximation of by . We assume that is a sequence of semimartingales satisfying the so-called condition (UT), and we prove that if converges weakly to in the topology, then converges weakly in the topology to . Moreover, we prove convergence of finite-dimensional distributions of to the corresponding finite-dimensional distributions of outside the set of discontinuity points of and . Consequently, using discrete approximations constructed in a manner analogous to Euler’s formula, we prove the existence of a weak solution of the SDE (1), provided that is continuous and satisfies (2). Moreover, if the SDE (1) has the weak uniqueness property, then our approximations computed by simple recurrent formulas allows us to obtain numerical solution of the SDE (1). In the case of reflected diffusion processes, similar approximation schemes have been considered previously (see, e.g., Liu [18], Pettersson [23], Słomiński [26]). In this section we also present some natural conditions ensuring convergence of to in probability provided that (1) has the so-called pathwise uniqueness property. Related results concerning diffusion processes have been given by, for instance, Kaneko and Nakao [12], Gyöngy and Krylov [6], Bahlali, Mezerdi and Ouknine [3], Alibert and Bahlali [1] and Słomiński [26].

We note that we consider the space equipped with two different topologies,  and . Definitions and required results for the Skorokhod topology have been given by, for example, Billingsley [4] and Jacod and Shiryayev [8]. For the convenience of the reader, we have collected basic definitions and properties of the -topology in the Appendix. More details have been provided in Jakubowski [10].

In this paper, we use the following notation. Every process appearing in the sequel is assumed to have trajectories in the space . If is a semimartingale, then represents and represents the quadratic variation process of , . Similarly, , and represents the predictable compensator of , . If is the process with locally finite variation, then , where is a total variation of on . In general, we let and denote convergence in law and in probability, respectively. To avoid ambiguity, we write () in if converges weakly (in probability) to in the space equipped with . Following [10], we write () in when we consider the topology. For , , , we let and denote classical moduli of continuity of on , that is, , and where . We also use the modulus introduced in Jakubowski [10]. We recall that for , , .

2 A deterministic case

Let be a nonempty convex (possibly unbounded) open set in , and let denote the set of inward normal unit vectors at ( if and only if for every , where denotes the usual inner product in ). The following remark also can be found in Menaldi [20] or Storm [27].

{remark}

(i) If , then there exists a unique such that . Moreover, .

(ii) For every ,

where .

Let be a function with initial value in . We recall that a pair of functions is called a solution of the Skorokhod problem associated with if

  • ,

  • is -valued,

  • is a function with locally bounded variation such that and

    where if .

The problem of existence of solutions of the Skorokhod problem and its approximation by solutions of equations with penalization terms has been discussed by many authors. Tanaka [29] proved existence and uniqueness of solutions in the case of continuous and domains also satisfying the following condition:

    []
  1. there exist constants and such that for every , we can find such that and .

Tanaka also observed that holds true in dimensions 1 and 2 or if is a bounded set. On the other hand, in dimension 2, one can construct examples of nonbounded convex domains not satisfying (). For instance, the cone with the basis and peak at , that is, the set

(6)

does not satisfy (). Cépa [5] omitted the assumption and proved the existence and uniqueness of the solution to the Skorokhod problem in the case of continuous function . In addition, Cépa proved convergence , of solutions of equations (4) for every sequence such that , .

The case of functions with jumps was considered for the first time by Anulova and Liptser [2], who proved the existence and uniqueness of solutions under condition (). Their result was generalized to the case of arbitrary convex by Łaukajtys [14]. In an earlier work [15], we considered the problem of approximating noncontinuous by solutions of equations with penalization terms only in this very special case. We now consider the problem of approximating noncontinuous by solutions of equations with penalization terms in the general case of arbitrary sequences such that in . Our main tools are the following estimates on the solution of (4):

Lemma \theremark

Let , and let be a solution of the equation (4). Then for any and such that

(7)

we have {longlist} (i) (ii) where , and denotes the largest integer less or equal to .

{pf}

We follow the proof of Theorem 3.2 in [5]. Let . Because is a continuous function such that ,

Therefore, for any ,

By Remark 2(ii),

and, consequently,

By (7), there exists a subdivision of such that , , , where and . Thus, in particular,

Therefore,

which implies that

(8)

From (8), it follows immediately that

Set . Then

which implies that

Thus, , and the proof of (i) is complete.

(ii) Using (8) and (i) gives

for . Thus, , which completes the proof.

Theorem \theremark

Assume that , , and let denote the solution of the equation (4), . If in , then {longlist} (i) and , , (ii) , provided that , (iii) , where denotes the solution of the Skorokhod problem associated with .

{pf}

(i) Because is relatively compact in , for any , and for any , , there exists such that . Therefore, the first conclusion follows from Lemma 2.

(ii) Let be a sequence of constants such that and . Set , , , where is an array of constants satisfying and . Now, for every , set , , , , . Observe that for every ,

Let be a solution of an equation with a penalization term of the form

Fix and consider the decomposition , where denotes a solution of the Skorokhod problem associated with . Due to [15], Lemma 2.2(i),

where variations , are bounded uniformly by Lemma 2(ii). Therefore,

Moreover, if , then . Because, by [15], Lemma 3.3, , , it follows that . On the other hand, by [29], Lemma 2.2, , and (ii) follows.

(iii) The sequence is relatively compact in , and consequently it is relatively compact. Because by part (i), , , the sequence is also relatively compact, and thus is relatively compact as well. In view of Corollary Appendix: The topology (Appendix), this proves (iii).

Recall that if in , then for every there exists a sequence such that

(9)

Moreover, for arbitrary sequences , such that , , and , we have

(10)

(see, e.g., [8], Chapter VI, Proposition 2.1).

Corollary \theremark

Under the assumptions of Theorem 2, {longlist} (i) for every , if is a sequence satisfying (9), then and for arbitrary sequences , such that , and , we have (ii) (iii) moreover, if is continuous, then

{pf}

(i) If , are step functions, then the result follows from [15], Lemma 3.3. In the general case, it is sufficient to use (9), (10) and repeat the approximation procedure from the proof of Theorem 2.

(ii) By Theorem 2(ii), , provided that . Therefore, it suffices to prove that is relatively compact in . Because , , , it is sufficient to show that

(11)

and

(12)

To prove (11), first observe that and , which implies that (11) is equivalent to the following:

  • for every sequence such that ,

    (13)

Next, note that (13) is implied by (i) (it is sufficient to put and observe that in this case, ).

Similarly, (12) is equivalent to the condition

  • for every and every sequence , such that , , if , , , , then

    (14)

Because is continuous and , it follows from (i) that for arbitrary sequences , such that and , we have