Equivalence of Uniqueness in Law and Joint Uniqueness in Law for SDEs Driven by Poisson Processes

We give an extension result of Watanabe’s characterization for 2-dimensional Poisson processes. By using this result, the equivalence of uniqueness in law and joint uniqueness in law is proved for one-dimensional stochastic differential equations driven by Poisson processes. After that, we give a simplified Engelbert theorem for the stochastic differential equations of this type.


Introduction
There are several types of solutions and uniqueness for stochastic differential equations, such as strong solution, weak solution, pathwise uniqueness, uniqueness in law and joint uniqueness in law, which will be introduced in Section 2. The relationship between them was firstly studied by Yamada and Watanabe [1] which is the famous Yamada-Watanabe theorem.It's an important method to prove the existence of strong solution for SDEs Nowadays.The study on this topic is still alive today and new papers are published, see [2]- [10].On the other hand, Jacod [11] and Engelbert [12] extended the Yamada-Watanabe theorem to the when the constrains are simple (linear) equations.It's sad that the stochastic differential equations are not of the form above, therefore the equivalence does not follow from this result.There exist few results for this question.As far as we know, Cherny [14] and Brossard [13] proved the equivalence of uniqueness in law and joint uniqueness in law for Itô equations of the following type driven by Brownian motion with the coefficients which only need to be measurable.Later, Qiao [15] extended the result of [14] to a type of infinite dimensional stochastic differential equaion.For stochastic differential equations with jumps, there is still no such result.So, in this paper, we are concerned with the following onedimensional stochastic differential equation driven by Poisson process ( ) ( ) We will give an extension form of Watanabe's characterization for 2-dimensional Poisson process, then by applying Cherny's approach, we prove the equivalence of the uniqueness in law and joint uniqueness in law for Equation (1).
This paper is organized as follows.In Section 2, we prepapre some notations and some definitions.After that, the main results are given and proved in Section 3.

Notations and Definitions
Let ( ) be the space of all càdlàg functions: + →   and let ( ) denote the σ-algebra generated by all the maps : We have the following Watanabe characterization for one dimensional Poisson process (see [16]).

Lemma 2.3. Let ( )
, ,P Ω  be a probability space with a given filtration . Assume that N is a counting process and that t t λ → is a deterministic function.Assume furthermore that the process M, defined by Then N is a F-Poisson process with intensity function λ .
In this paper, we consider the following stochastic differential equation driven by the Poisson process ( ) ( ) where : and N is a Poisson process with intensity function λ on a stochastic basis Law X Law X ′ =  Definition 2.6.We say that joint uniqueness in law holds for (2.1) if whenever ( ) , X N and ( )

Main Results
Theorem 3.1.Suppose that the uniqueness in law holds for (2.1).Then, for any solutions ( ) ( ) , X N and the law of ( ) According to Theorem 1.5 of Kurtz [7], we have the following simplified Yamada-Watanabe-Engelbert theorem (see aslo [12] , :


, which tell us that N is a counting process.Furthermore, we have process defined by ( ) We will recall the concept of conditional distribution from the measure theory.Let : E ξ Ω → be a random element on ( ) . Let ⊆   , then there exists a conditional distribution of ξ with respect to  , that is, a family ( ) ) , the map 2) For any ( ) The conditional distribution defined above is unique in the sense: if ( ) family probability measures with the same properties, then Q Q ω ω =  for P-a.e. ω .

2) If
( ) Lemma 3.5.Let ( ) , X N be a weak solution of (2.1) on a filtered probability space ( ) ( ) , , , t P Ω   .Let ( ) Q ω ω∈Ω be a conditional distribution of ( ) , X N with respect to 0  (here we consider ( ) , X N as a ×   -valued random variable).We denote by Y, M the canonical maps from ×   onto  respectively, that is .
Then, for P-a.e. ω , the pair ( ) Proof.Let us check the conditions of Definition 2.4.
1) Firstly, we will check that M is an ( ) Λ is defined as in Definition 2.1.Hence, we have We deduce that, for P-a.e ω , M is an ( ) -Poisson process with intensity function λ .
. They got Pathwise uniqueness Uniqueness in law ⇒ and Weak solution Pathwise uniqueness Strong solution, + ⇒

and 2 S
stochastic differential equation driven by semi-martingales.Especially, Engelbert got an inverse result, that is Strong solution Joint uniqueness in law Pathwise uniqueness, + ⇒ which can be seen as a complement of the Yamada-Watanabe theorem.Recently, Kurtz [5] [7] considered an abstract stochastic equation of the form are Polish spaces.They obtained an unified result ([7] Theorem 1.5): Strong solution Joint uniqueness in law Weak solution Pathwise uniqueness + + which was called the Yamada-Watanabe-Engelbert thereom.This result can cover most results mentioned above.However, joint uniqueness in law is harder to check than uniqueness in law in view of application.The natural question that arises now is: under what conditions, joint uniqueness can be equivalent to uniqueness in law?Kurtz ([5] [7]) gave a positive answer for the stochastic equations of the form
Definition 2.5.We say that uniqueness in law holds for (2.1) if whenever(  ), X N and ( ) Definition 2.7.We say that pathwise uniqueness holds for (1.1) if whenever ( ) Theorem 3,[14]Theorem 3.2) immediately.We have the following generalised martingale characterization for 2-dimensional Poisson processes, which may have its own interest.
, ,P Ω  be a probability space with a given filtration By Lemma 2.3, we only need to prove that two Poisson processes are independent if and only if their sum is also a Poisson process.
Then, N is a 2-dimensional F-Poisson process with intensity function ( ) e.. N can be defined on Ω  in an obvious way.The pair ( )