Uniqueness Of Strong Solutions To Stochastic Differential Equations

Yamada-Watanabe's Theorem on the existence and uniqueness of solutions of stochastic differential equation driven by Brownian motion plays a fundamental role in stochastic differential equation theory.It clarifies the relation between strong solution and weak solution.Cherny[1]gives a theorem which is somehow "dual" to the Yamada-Watanabe's Theorem.That is,the uniqueness in law for stochastic differential equations,together with the strong existence guarantees the pathwise uniqueness. Researchers have increasingly been studying models from economics and from natural sciences where the underlying randomness contains jumps.As we know,the Brownian motion and poisson process are special cases of Lévy process.Chen[2]tries to generalize Yamada-Watanabe's result to stochastic differential equations driven by Lévy process,and finally obtain a result comparable to the Yamada-Watanabe ont.In conjunction with the idea of Cherny[1]and Chen[2],we try to prove that:the pathwise uniqueness holds for stochastic differential equations driven by Lévy process provided that we have the uniqueness in law and the strong existence.Yeh[3]adopts Ikeda and Watanabe's approach in[4]and generalizes Yamada-Watanabe's result to stochastic differential equation in the plane,that is:under the assumption of the existence of a weak solution and the pathwise uniqueness of solutions,existence and uniqueness of a strong solution to stochastic differential system of non Markovian type in the plane is obtained where the boundary process is a continuous real valued function on(?)R_+~2.Considering the idea of Cherny[1],we prove that the existence and uniqueness of a strong solution to the former stochastic differential equation is obtained provided that we have the existence of a strong solution and the uniqueness in law.This result is somehow "dual" to the result of Yeh.The proof of the result based on a statement of interest in itself:if there is uniqueness in law for the stochastic differential equations of non-Markovian type in the plane,then the joint distribution Law(X(z),B(z);z∈R_+~2) is the same for all solutions(X,B).