Yamada-Watanabe Theorem for Stochastic Evolution Equation Driven by Poisson Random Measure

Abstract

The purpose of this paper is to give a detailed proof of Yamada-Watanabe theorem for stochastic evolution equation driven by pure Poisson random measure.

1. Introduction

The main purpose of this paper is to establish the Yamada-Watanabe theory of uniqueness and existence of solutions of stochastic evolution equation driven by pure Poisson random measure in the variational approach. The classical paper [1] has initiated a comprehensive study of relations between different types of uniqueness and existence (e.g., strong solutions, weak solutions, pathwise uniqueness, uniqueness, and joint uniqueness in law) arising in the study of SDEs (see, e.g., [2–4]) and the study is still alive today. New papers are published (see, e.g., [2, 3, 5–7]). In this paper we are concerned with the similar question for stochastic evolution equation driven by Poisson random measure by using the method of Yamada and Watanabe.

Yamada and Watanabe's initial work [1] proved that weak existence and pathwise uniqueness imply strong existence and weak uniqueness. For -dimensional case, see [8, 9]. For infinite dimensional stochastic differential equation, Ondreját [6] proved similar result for stochastic evolution equation in Banach space driven by cylindrical Wiener process, where the solutions are understood in the mild sense. Lately, Röckner et al. [7] proved similar result for stochastic evolution equation in Banach space driven by cylindrical Wiener process under the variational framework. On the other hand, Kurtz [2, 3] obtained a pleasant version of Yamada-Watanabe and Engelbert theorem in an abstract form, which covered most of the work mentioned above. However, we will consider the following concrete stochastic evolution equation by using a different method.

In this paper, we will consider the following stochastic evolution equation driven by pure Poisson random measure under the variational framework: This type of equations can be applied to many SPDEs, for example, stochastic Burgers equation, stochastic porous media equation, and stochastic Navier-Stokes equation (see, e.g., [9–13]). We will introduce the above equation precisely in Section 2. Our aim is to obtain this jump-case Yamada-Watanabe theorem; that is, weak existence and strong uniqueness (which will be stated in Section 2) imply strong existence and weak uniqueness and vice versa. We note that there are some differences between the jump-case case and the Brownian motion case. It is well known that a Brownian motion can be treated as a canonical map on or (for some Hilbert space ), while for jump-case we have to work on the configuration space (see Section 2) for Poisson random measure.

The structure of the present paper is as follows. In Section 2 we introduce the framework and definitions of strong solution and weak solution. In Section 3 we give and prove our main results.

2. Framework and Definitions

Let be a separable Hilbert space with inner product and norm . Let and be separable Banach space with norms and , such that continuously and densely.

Let be the space of all càdlàg functions: . Set if ; we extend to a function on . Note that this extension is -measurable and lower semicontinuous. Hence, the following path space is well defined: equipped with the metric where is the Skorokhod distance on ; see [14]. It is easy to see that is a Polish space. Let denote the -algebra generated by all the maps , , where , .

Let and let be a measurable space. We denote by the space of all -valued measures on and consider the measurable space , where is the -algebra generated by the mappings , .

Let be a -finite measure on . We recall that a Poisson random measure on with intensity is a measurable mapping from a probability space to such that the following two conditions hold:(i)for every , the random variable has the Poisson distribution with the mean ;(ii)for any disjoint , the random variables are independent.

Let be the distribution of ; then is a probability space, and the canonical map is a Poisson random measure with intensity on this probability space. Let . Note that is a Polish space (cf. [15]).

In this paper, let , where is an arbitrary locally compact Hausdorff topological space with countable base, and , where is a -finite measure on . We denote by the -algebra generated by the mappings: , for all , for all .

Let and be and -measurable, respectively, such that for each and

Let ; we consider the following stochastic evolution equation: with . For more results about this type of equation see [9, 11, 13] and the references therein.

Definition 1. Assume that is a filtered probability space and is a Poisson random measure defined as above. Then is called a -Poisson random measure with intensity if,(i)for any , with , then is -measurable;(ii)for any , with , then is independent of .

Remark 2. The canonical Poisson random measure on is -Poisson random measure.

Definition 3. For some , a pair , where is an -adapted process with paths in and is a -Poisson random measure with intensity on a stochastic basis , is called a weak solution of (7) if(i)for any (ii)as a stochastic equation on one has

Definition 4. One says that weak uniqueness holds for (7) if whenever and are two weak solutions with stochastic bases and such that then

Definition 5. One says that pathwise uniqueness holds for (7) if whenever and are two weak solutions on the same stochatic bases such that -a.s.; then -a.s.:

In order to define strong solutions one needs to introduce the following class of maps. Let denote the set of all maps such that for every probability measure on there exists a -measurable map such that for -a.e. Here denotes the completion of with respect to , where denotes the distribution of the Poisson random measure on space . Of course, is uniquely determined by -a.e.

Definition 6. A weak solution to (7) on is said to be a strong solution if there exists with respect to such that, for , is -measurable for every and where denote the completion with respect to in .

Definition 7. Equation (7) is said to have a unique strong solution, if there exists satisfying the condition in the above definition such that the following conditions hold. (1)For every Poisson random measure with intensity on a stochastic basis and any -measurable the càdlàg process is -adapted and satisfies (i) and (ii) of Definition 3; that is, is a weak solution to (7), and   -a.e.(2)For any weak solution to (7) one has

3. The Main Result and Its Proof

Let us now state the main result.

Theorem 8. Let and be as above. Then (7) has a unique strong solution if and only if both of the following properties hold. (i)For every probability measure on there exists a weak solution of (7) such that is the distribution of .(ii)Pathwise uniqueness holds for (7).

Proof. Suppose that (7) has a unique strong solution. Then (ii) obviously holds. To show (i) we only have to take the probability space and consider with filtration where denotes all -zero sets in . Let and : be the canonical projections. Then is the desired weak solution in (i).
Now let us suppose that (i) and (ii) hold. We are going to show that there exists a unique strong solution for (7).
Let with stochastic basis be a weak solution to (7) with initial distribution . Define a probability measure on , by We have the following lemma.

Lemma 9. There exists a family , , , of probability measures on having the following properties.(i)For every the map is -measurable.(ii)For every -measurable function , one has (iii)If and is -measurable, then is -measurable, where denotes the completion with respect to in .

Proof. Let be the canonical projection. Since is -measurable, hence -independent of , it follows that We recall that is a Polish space and, by the existence result on regular conditional distributions the family , , , exist and satisfy (i) and (ii).
To prove (iii) it suffices to show that for and for all , , , and Since the system of all , , as above generate . According to (ii) above, the left side of (24) is equal to Note that is a Poisson random measure on with stochastic basis , so is -independent of ; then the right side of (25) is since is -independent of .

For define a measure on by

Define the stochastic basis where and define the maps

Then, it is easy to see that

Lemma 10. There exists with such that, for all , one has that is an -Poisson random measure on .

Proof. By definition is -adapted for every . We only need to check that condition (ii) of Definition 1 holds for almost all . Furthermore, for , and , , , , , which ends the proof.

Fix a probability measure on and let with stochastic basis be a weak solution to (7) with initial distribution . Let be a predictable process defined as above. In particular we assume that, for some , -a.e. for all . We aim to give a pointwise definition for stochastic integral . First of all, we give a pointwise definition for stochastic integral at the fixed time .

For , define ; it is easy to see that for . Also note that pointwise on where the infimum of the empty set is taken to be infinity. Firstly, let be the elementary predictable process and define the stochastic integral as usual (cf. [12, page 130]). That is, if , where , such that , and are bounded and -measurable, then Fix and let be the elementary predictable such that as . By Define where the limit exists in and almost everywhere; then put

Now we map back to the stochastic basis ; we find that is a sequence of -stopping times with   -a.e. Also are -predictable processes that converge in to , so that converge in to the stochastic integral . Putting these pieces together shows that is a version of the stochastic integral .

By the same argument as in [7], we have the following two lemmas.

Lemma 11. There exists with such that, for , is a -version of the stochastic integral on the stochastic basis .

Lemma 12. There exists with such that for all , and with stochastic basis are weak solutions of (7) such that therefore, -a.e.

Lemma 13. There exists a -measurable map such that for -a.e. where is the Dirac measure on with mass in . Furthermore, is -measurable for all .

Proof. By Lemma 12, for all , we have where . Note that is Polish space, by Lemma 2.2 in [7], there exists such that and for all there exists such that Set if . Let ; then and the other properties of follow from Lemma 9.

Having defined the mapping let us check the conditions of Definitions 6 and 7.

Lemma 14. One has

Proof. By Lemmas 9 and 13, we have

Let be another -Poisson random measure on with intensity on a stochastic basis and let be an -measurable map and . Let be as above and set By the same argument as in [7], we also have the following lemma.

Lemma 15. is a weak solution to (7) with   -a.e.

To complete the proof we still have to construct and to check the adaptiveness conditions on it. Below we will apply what we have obtained above to replacing . So, for each , we have a function . Now, define

The proof is complete by the following lemma.

Lemma 16. Let be a probability on and as constructed in Lemma 13. Then for -a.e. Furthermore, is -measurable for every , .

Proof. Let and fix . Define a measure on by where is as in Lemma 9. Consider the stochastic basis , where where Let As above, we can show that on is a weak solution to (7) with -a.e., and on is also a weak solution to (7) with Hence, by our pathwise uniqueness assumption (ii), we have Hence, for and by Lemma 13 and (56) But for each Since is -measurable, so is the right side of (58). Integrating the above equation with respect to and we obtain which proofs the assertion.
Let , , , and define Then hence, But so by (62) it follows that

Remark 17. Our result can be applied to paper [10, page 19] to make sure that there exists a measurable map such that is the unique strong solution of the equation they considered.

4. Conclusions

What we have seen from above is a Yamada-Watanabe theorem for a type of stochastic partial differential equations with jumps, proved by using the original idea of Yamada and Watanabe’s. Comparing with results of Kurtz [2, 3], their results can also imply the same result, where they used another abstract method.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the three anonymous referees for useful comments and suggestions. The author is also indebted to Professor Jiagang Ren for his valuable discussion and for the guidance over the past years. This work was supported by the Young Teacher Research Program of BNUZ (Grant no. 201353051).