On Weak Solutions of Backward Stochastic Differential Equations

Abstract

The main objective of this paper consists in discussing the concept of weak solutions of a certain type of backward stochastic differential equations. Using weak convergence in the Meyer--Zheng topology, we shall give a general existence result. The terminal condition H depends in functional form on a driving càdlàg process X, and the coefficient f depends on time t and in functional form on X and the solution process Y. The functional $f(t,x,y),(t,x,y)\in [0,T]\times D([0,T];{\bf R}^{d+m})$ is assumed to be bounded and continuous in $(x,y)$ on the Skorokhod space $D([0,T]\,;{\bf R}^{d+m})$ in the Meyer--Zheng topology. By several examples of Tsirelson type, we will show that there are, indeed, weak solutions which are not strong, i.e., are not solutions in the usual sense. We will also discuss pathwise uniqueness and uniqueness in law of the solution and conclude, similar to the Yamada--Watanabe theorem, that pathwise uniqueness and weak existence ensure the existence of a (uniquely determined) strong solution. Applying these concepts, we are able to state the existence of a (unique) strong solution if, additionally to the assumptions described above, f satisfies a certain generalized Lipschitz-type condition.