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 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 2, Pages 390–396 (Mi tvp117)

Short Communications

A backward stochastic differential equation without strong solution

R. Buckdahna, H. J. Engelbertb

a Université de Bretagne Occidentale
b Friedrich-Schiller-University

Abstract: In [R. Buckdahn, H.-J. Engelbert, and A. Răşcanu, 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.

Keywords: backward stochastic differential equations, weak solutions, strong solutions, Tsirelson example.

DOI: https://doi.org/10.4213/tvp117

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English version:
Theory of Probability and its Applications, 2006, 50:2, 284–289

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Citation: R. Buckdahn, H. J. Engelbert, “A backward stochastic differential equation without strong solution”, Teor. Veroyatnost. i Primenen., 50:2 (2005), 390–396; Theory Probab. Appl., 50:2 (2006), 284–289

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp117
• https://doi.org/10.4213/tvp117
• http://mi.mathnet.ru/eng/tvp/v50/i2/p390

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This publication is cited in the following articles:
1. Theory Probab. Appl., 52:1 (2008), 152–160
2. 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
3. Ma J., Zhang J., “On weak solutions of forward-backward SDEs”, Probab Theory Related Fields, 151:3–4 (2011), 475–507
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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