On Weak Solutions of Backward Stochastic Differential Equations

Preprint series: 03-14, Reports on Stochastics and Statistics

Buckdahn, R., H.-J. Engelbert, A. Rascanu

MSC:
60H20 Stochastic integral 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 c\{a}dl\{a}g process $X$, and the coefficient
$f$ depends on time $t$ and in functional form on $X$ and the solution process
$Y$. The functional $f(t,x,y),(t,x,y)\in\lbrack0,T]\times D\left( [0,T];R^{d+m}\right)$, is assumed to be bounded and continuous in $(x,y)$ on
the Skorohod space $D\left( [0,T];R^{d+m}\right)$ 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.

Keywords: Backward stochastic differential equations, weak solutions, strong solutions, pathwise uniqueness, uniqueness in law, weak convergence, Meyer-Zheng topology