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

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

**MSC:**- 60H10 Stochastic ordinary differential equations [See also 34F05]
- 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*

**Upload:** 2003-11-12

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