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 Teor. Veroyatnost. i Primenen., 2007, Volume 52, Issue 1, Pages 190–199 (Mi tvp15)

This article is cited in 6 scientific papers (total in 6 papers)

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On the continuity of weak solutions of backward stochastic differential equations

R. Buckdahna, H.-J. Engelbertb

a Laboratoire des Mathématiques, Université de Bretagne Occidentale, Brest, France
b Institut für Stochastik, Friedrich Schiller-Universität, Jena, Germany

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. 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.

DOI: https://doi.org/10.4213/tvp15

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English version:
Theory of Probability and its Applications, 2008, 52:1, 152–160

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Received: 07.09.2006
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Citation: R. Buckdahn, H.-J. Engelbert, “On the continuity of weak solutions of backward stochastic differential equations”, Teor. Veroyatnost. i Primenen., 52:1 (2007), 190–199; Theory Probab. Appl., 52:1 (2008), 152–160

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Yannacopoulos A.N., Frangos N.E., Karatzas I., “Wiener chaos solutions for linear backward stochastic evolution equations”, SIAM J. Math. Anal., 43:1 (2011), 68–113
2. Ma J., Zhang J., “On weak solutions of forward-backward SDEs”, Probab. Theory Related Fields, 151:3-4 (2011), 475–507
3. Liang G., Lyons T., Qian Zh., “Backward stochastic dynamics on a filtered probability space”, Ann. Probab., 39:4 (2011), 1422–1448
4. Bouchemella N. de Fitte P.R., “Weak Solutions of Backward Stochastic Differential Equations with Continuous Generator”, Stoch. Process. Their Appl., 124:1 (2014), 927–960
5. Carmona R. Delarue F., “Probabilistic Theory of Mean Field Games With Applications i: Mean Field Fbsdes, Control, and Games”, Probabilistic Theory of Mean Field Games With Applications i: Mean Field Fbsdes, Control, and Games, Probability Theory and Stochastic Modelling, 83, Springer International Publishing Ag, 2018, 1–713
6. Carmona R. Delarue F., “Probabilistic Theory of Mean Field Games With Applications II: Mean Field Games With Common Noise and Master Equations”, Probabilistic Theory of Mean Field Games With Applications II: Mean Field Games With Common Noise and Master Equations, Probability Theory and Stochastic Modelling, 84, Springer International Publishing Ag, 2018, 1–697
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