Uniqueness of solutions for multidimensional BSDEs with uniformly continuous generators

ShengJun Fan; Long Jiang; Matt Davison

doi:10.1016/j.crma.2010.03.013

Probability Theory

Uniqueness of solutions for multidimensional BSDEs with uniformly continuous generators

Presented by: Paul Malliavin

ShengJun Fan ¹; Long Jiang ¹; Matt Davison ²

¹ College of Sciences, China University of Mining & Technology, Xuzhou, Jiangsu 221116, PR China
² Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7

Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 683-686.

Abstract
Résumé

Hamadène (2003) [2] obtained an existence result of solutions for multidimensional backward stochastic differential equations (BSDEs) with uniformly continuous generators, provided that the ith component $g_{i} (t, y, z)$ of the generator g depends only on the ith row of the matrix z. The uniqueness of solutions for this kind of BSDEs is proved in this Note.

Hamadène (2003) [2] a obtenu un résultat d'existence des solutions d'équations différentielles stochastiques rétrogrades (BSDE) à générateurs uniformément continus à condition que la i-ème composante $g_{i} (t, y, z)$ du générateur g ne dépende que de la i-ème ligne de la matrice z. L'unicité de la solution pour ce type d'équations BSDE est démontrée dans cette Note.

Received: 2010-01-28
Accepted: 2010-03-18
Published online: 2010-05-23

DOI: 10.1016/j.crma.2010.03.013

Author's affiliations:

ShengJun Fan ¹; Long Jiang ¹; Matt Davison ²

¹ College of Sciences, China University of Mining & Technology, Xuzhou, Jiangsu 221116, PR China
² Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7

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     author = {ShengJun Fan and Long Jiang and Matt Davison},
     title = {Uniqueness of solutions for multidimensional {BSDEs} with uniformly continuous generators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {683--686},
     publisher = {Elsevier},
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     year = {2010},
     doi = {10.1016/j.crma.2010.03.013},
     language = {en},
}

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ShengJun Fan; Long Jiang; Matt Davison. Uniqueness of solutions for multidimensional BSDEs with uniformly continuous generators. Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 683-686. doi : 10.1016/j.crma.2010.03.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.013/

References
Cited by

[1] S. Fan; L. Jiang Uniqueness result for the BSDE whose generator is monotonic in y and uniformly continuous in z, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010), pp. 89-92

[2] S. Hamadène Multidimensional backward stochastic differential equations with uniformly continuous coefficients, Bernoulli, Volume 9 (2003) no. 3, pp. 517-534

[3] X. Mao Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients, Stochastic Process and Their Applications, Volume 58 (1995), pp. 281-292

[4] E. Pardoux BSDEs, weak convergence and homogenization of semilinear PDEs, Montreal, QC, 1998, Kluwer Academic Publishers, Dordrecht (1999), pp. 503-549

[5] E. Pardoux; S. Peng Adapted solution of a backward stochastic differential equation, Systems Control Letters, Volume 14 (1990), pp. 55-61

Cited by Sources:

^☆ Supported by the National Natural Science Foundation of China (No. 10971220), the FANEDD (No. 200919), the National Basic Research Program of China (No. 2007CB814901) and Qing Lan Project.

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