A Backward Stochastic Differential Equation without Strong Solution

Abstract

In [R. Buckdahn, H.-J. Engelbert, and A. R\u{a}\c{s}canu, Theory Probab. Appl., 49 (2005), pp. 16-50] the notion of a weak solution of a general backward stochastic differential equation (BSDE) was introduced. There was also given an example of a weak solution for a certain BSDE which is not a strong solution, i.e., not a solution in the classical sense. However, the solution of the BSDE which was considered is not unique in law and, as was pointed out, there exist also strong solutions of this BSDE. In the present paper, we will remove this insufficiency and give an example of a BSDE which has a weak solution but does not possess any strong solution.