Received 23 November 2015; accepted 17 May 2016; published 20 May 2016
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  . They got
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  -  . On the other hand, Jacod  and Engelbert  extended the Yamada-Watanabe theorem to the stochastic differential equation driven by semi-martingales. Especially, Engelbert got an inverse result, that is
where, and are Polish spaces. They obtained an unified result (  Theorem 1.5):
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 (   ) gave a positive answer for the stochastic equations of the form
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  and Brossard  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  extended the result of  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 one- dimensional 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.
2. Notations and Definitions
Let be the space of all càdlàg functions: and let denote the s-algebra generated by all the maps, , where,. Let.
Definition 2.1. Let be a probability space with a given filtration, and let be a deterministic function of time. A counting process N is a Poisson process with intensity function with respect to the filtration if it satisfies the following conditions.
1) N is adapted to;
2) For all the random variable is independent of;
3) For all, the conditional distribution of the increment is given by
Definition 2.2. Let be a probability space with a given filtration, and let, be two deterministic function of time. and are two F-Poisson processes with intensity function and respectively. Process is called a 2-dimensional F-Poisson process with intensity function if and are independent.
We have the following Watanabe characterization for one dimensional Poisson process (see  ).
Lemma 2.3. Let be a probability space with a given filtration. Assume that N is a counting process and that is a deterministic function. Assume furthermore that the process M, defined by
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 and are―measurable and for each, and are predictable.
Definition 2.4. A pair, where is a càdlàg -adapted process with paths in and N is a Poisson process with intensity function on a stochastic basis, is called a weak solution of (2.1) if
1) For all,
2) For all,
Definition 2.5. We say that uniqueness in law holds for (2.1) if whenever and are two weak solutions with stochatic bases and such that
Definition 2.6. We say that joint uniqueness in law holds for (2.1) if whenever and are two weak solutions with stochatic bases and such that
Definition 2.7. We say that pathwise uniqueness holds for (1.1) if whenever and are two weak solutions on the same stochatic bases such that P-a.s., then P-a.s.
3. Main Results
Theorem 3.1. Suppose that the uniqueness in law holds for (2.1). Then, for any solutions and, the law of and the law of are equal on, that is
According to Theorem 1.5 of Kurtz  , we have the following simplified Yamada-Watanabe-Engelbert theorem (see aslo  Theorem 3,  Theorem 3.2) immediately.
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 be a probability space with a given filtration. Assume that is a 2-dimensional counting process and that are two deterministic function. Then, N is a 2-dimensional F-Poisson process with intensity function is equivalent to the following two conditions.
1) Processes and defined by
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 and are two independent Poisson processes. For, we have,
By the independence of and, for each, we obtain
We conclude that, which tell us that N is a counting process. Furthermore, we have process defined by
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 be a Poisson process, we aim to prove that and are independent. In fact, let and, , we have
which completes the proof.
We will recall the concept of conditional distribution from the measure theory. Let be a random element on taking value in a Polish space. Let, then there exists a conditional distribution of with respect to, that is, a family of probability measures on such that
1) For any, the map is -measurable;
2) For any, ,
Remark 3.4. 1) The conditional distribution defined above is unique in the sense: if is another family probability measures with the same properties, then for P-a.e..
2) If is such that, then for P-a.e..
Lemma 3.5. Let be a weak solution of (2.1) on a filtered probability space. Let be a conditional distribution of with respect to (here we consider as a -valued random variable). We denote by Y, M the canonical maps from onto respectively, that is
Then, for P-a.e., the pair is a weak solution of (2.1) on.
Proof. Let us check the conditions of Definition 2.4.
1) Firstly, we will check that M is an -Poisson process. For any, , , we have
where is defined as in Definition 2.1. Hence, we have
It follows that
Therefore, for P-a.e.,
We deduce that, for P-a.e, M is an -Poisson process with intensity function.
2) For any,
By Remark 3.4, we have
3) We have
Proof of Theorem 3.1. Let be a weak solution of (2.1) on a filtered space. Let and be two independent -Poisson processes with the same intensity function. Set
Then X, N, and can be defined on in an obvious way. The pair is a solution of (2.1) on and, are independent -Poisson processes. For any, and, is a linear operator from. Let denote the orthogonal projection from; let denote the orthogonal projection from.
For any, set
We claim that and are two independent -Poisson processes with the intensity function. In fact, it’s easy to see that, and are counting processes. Moreover, Let
Note that processes and are predictable precesses and the integrators in the above equations are martingales. We conclude that and are martingale, theorefore is also a martingale. By Lemma 2.3, we get that and are two―Poisson processes with the intensity function and is a―Poisson processes with the intensity function. By Lemma 3.3, we deduce that and are two independent -Poisson processes.
For any, we have
Consequently, is also a solution of (2.1) on.
Let us now consider the filtration
. Note that, for any, the s-fields and are indepen- dent. Thus, is a -Poisson process. So, the pair is also a solution of (2.1) on.
Let be a conditional distribution of with respect to. By Lemma 3.5, is a solution of (2.1) on for -a.e.. As the uniqueness in law holds for (2.1), the distribution (which is the conditional distribution of X with respect to) is the same for -a.e.. This means that the process X is independent of. In particular, X and are independent.
For any and, let be the pseudo inverse of. It is easy to check that is predictable and we have. It follows that
By (3.1), we get
The process is a measurable functional of X, while is independent of X. Thus, the last equation shows that the distribution is unique. ,
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).