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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.

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 [

and

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 [

which can be seen as a complement of the Yamada-Watanabe theorem. Recently, Kurtz [

where

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 ( [

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 [

driven by Brownian motion with the coefficients which only need to be measurable. Later, Qiao [

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.

Let

Definition 2.1. Let

1) N is adapted to

2) For all

3) For all

where

Definition 2.2. Let

We have the following Watanabe characterization for one dimensional Poisson process (see [

Lemma 2.3. Let

is an F-martingale. Then N is a F-Poisson process with intensity function

In this paper, we consider the following stochastic differential equation driven by the Poisson process

where

Definition 2.4. A pair

1) For all

2) For all

Definition 2.5. We say that uniqueness in law holds for (2.1) if whenever

then

Definition 2.6. We say that joint uniqueness in law holds for (2.1) if whenever

then

Definition 2.7. We say that pathwise uniqueness holds for (1.1) if whenever

Theorem 3.1. Suppose that the uniqueness in law holds for (2.1). Then, for any solutions

According to Theorem 1.5 of Kurtz [

Corollary 3.2. The following are equivalent:

1) Equation (2.1) has a strong solution and uniqueness in law holds;

2) Equation (2.1) has a weak solution and pathwise uniqueness holds.

We have the following generalised martingale characterization for 2-dimensional Poisson processes, which may have its own interest.

Lemma 3.3. Let

1) Processes

are F-martingales.

2) Process N defined by

is a F―Poisson process with intensity function

Proof. 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.

Suppose that

By the independence of

We conclude that

is a martingale. By the Watanabe’s result, we have that N is a Poisson process with intensity function

On the other hand, suppose that

which completes the proof.

We will recall the concept of conditional distribution from the measure theory. Let

1) For any

2) For any

Remark 3.4. 1) The conditional distribution defined above is unique in the sense: if

2) If

Lemma 3.5. Let

and

Let

Then, for P-a.e.

Proof. Let us check the conditions of Definition 2.4.

1) Firstly, we will check that M is an

where

It follows that

Therefore, for P-a.e.

We deduce that, for P-a.e

2) For any

By Remark 3.4, we have

for P-a.e.

3) We have

Hence,

for P-a.e.

Proof of Theorem 3.1. Let

Then X, N,

For any

We claim that

We have

Note that processes

For any

Consequently,

Let us now consider the filtration

Let

For any

where

By (3.1), we get

The process

Remark 3.6. In this paper, the equivalence of the uniqueness in law and joint uniqueness in law holds when diffusion coefficient may be degenerate. We note that, for the general multidimensional stochastic differential equations with jumps, the equivalence does not hold when the diffusion coefficients are allowed to be degenerate. We will consider in the future study.

This work was supported in part by the National Natural Science Foundation of China(Grant No.11401029) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ13A010020).

Huiyan Zhao,Chunhua Hu,Siyan Xu, (2016) Equivalence of Uniqueness in Law and Joint Uniqueness in Law for SDEs Driven by Poisson Processes. Applied Mathematics,07,784-792. doi: 10.4236/am.2016.78070