Geodesic
On the continuity of weak solutions of backward stochastic differential equations
Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 1, pp. 190-199

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In the present paper, the notion of a weak solution of a general backward stochastic differential equation (BSDE), which was introduced by the authors and A. Rǎşcanu in [Theory Probab. Appl., 49 (2005), pp. 16–50], will be discussed. The relationship between continuity of solutions, pathwise uniqueness, uniqueness in law, and existence of a pathwise unique strong solution is investigated. The main result asserts that if all weak solutions of a BSDE are continuous, then the solution is pathwise unique. One should notice that this is a specific result for BSDEs and there is of course no counterpart for (forward) stochastic differential equations (SDEs). As a consequence, if a weak solution exists and all solutions are continuous, then there exists a pathwise unique solution and this solution is strong. Moreover, if the driving process is a continuous local martingale satisfying the previsible representation property, then the converse is also true. In other words, the existence of discontinuous solutions to a BSDE is a natural phenomenon, whenever pathwise uniqueness or, in particular, uniqueness in law is not satisfied. Examples of discontinuous solutions of a certain BSDE were already given in [R. Buckdahn and H.-J. Engelbert, Proceedings of the Fourth Colloquium on Backward Stochastic Differential Equations and Their Applications, to appear]. This was the motivation for the present paper which is aimed at exploring the general situation.
Keywords: backward stochastic differential equations, weak solutions, strong solutions, uniqueness in law, pathwise uniqueness, continuity of solutions, discontinuity of solutions. 
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     title = {On the continuity of weak solutions of backward stochastic differential equations},
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R. Buckdahn; H.-J. Engelbert. On the continuity of weak solutions of backward stochastic differential equations. Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 1, pp. 190-199. http://geodesic.mathdoc.fr/item/TVP_2007_52_1_a14/