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ON THE CONTINUITY OF WEAK SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS.
- Source :
- Theory of Probability & Its Applications; 2008, Vol. 52 Issue 1, p152-160, 9p
- Publication Year :
- 2008
-
Abstract
- 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. Răşcanu 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0040585X
- Volume :
- 52
- Issue :
- 1
- Database :
- Complementary Index
- Journal :
- Theory of Probability & Its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 31817134
- Full Text :
- https://doi.org/10.1137/S0040585X9798292X