Uniqueness result for the BSDE whose generator is monotonic in <i>y</i> and uniformly continuous in <i>z</i>

Sheng-Jun Fan; Long Jiang

doi:10.1016/j.crma.2009.10.023

Probability Theory

Uniqueness result for the BSDE whose generator is monotonic in y and uniformly continuous in z
[Un résultat d'unicité pour une équation différentielle stochastique rétrograde dont le générateur g est monotone en y et uniformément continue en z]

Présenté par : Paul Malliavin

Sheng-Jun Fan ¹ ; Long Jiang ¹

¹ College of Sciences, China University of Mining & Technology, Xuzhou, Jiangsu 221116, PR China

Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 89-92.

Abstract (VO)
Résumé

In this Note, we prove that if g is continuous, monotonic and has a general growth in y, g is uniformly continuous in z, and $(g {(t, 0, 0))}_{t \in [0, T]}$ is square integrable, then for each square integrable terminal condition ξ, the one-dimensional backward stochastic differential equation (BSDE) with the generator g has a unique solution. This generalizes some corresponding (one-dimensional) results.

Dans cette Note on démontre que si g est continue, monotone, de croissance quelconque en y, g uniformément continue en z et $(g {(t, 0, 0))}_{t \in [0, T]}$ est de carré intégrable, alors pour toute condition finale ξ de carré intégrable, en dimension un, l'équation différentielle stochastique rétrograde (BSDE) de générateur g, a une solution unique. Ce résultat généralise des résultats connus dans le cas de la dimension un.

Reçu le : 2009-01-24
Accepté le : 2009-10-06
Publié le : 2009-12-22

DOI : 10.1016/j.crma.2009.10.023

Affiliations des auteurs :

Sheng-Jun Fan ¹ ; Long Jiang ¹

¹ College of Sciences, China University of Mining & Technology, Xuzhou, Jiangsu 221116, PR China

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Sheng-Jun Fan; Long Jiang. Uniqueness result for the BSDE whose generator is monotonic in y and uniformly continuous in z. Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 89-92. doi : 10.1016/j.crma.2009.10.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.10.023/

Bibliographie
Cité par

[1] K. Bahlali Backward stochastic differential equations with locally Lipschitz coefficient, C. R. Acad. Sci. Paris, Ser. I, Volume 331 (2001), pp. 481-486

[2] Ph. Briand; B. Delyon; Y. Hu; E. Pardoux; L. Stoica $L^{p}$ solutions of backward stochastic differential equations, Stochastic Processes and Their Applications, Volume 108 (2003), pp. 109-129

[3] Ph. Briand; Y. Hu Quadratic BSDEs with convex generators and unbounded terminal conditions, Probability Theory and Related Fields, Volume 141 (2008), pp. 543-567

[4] Ph. Briand; J.P. Lepetier; J. San Martin One-dimensional BSDE's whose coefficient is monotonic in y and non-Lipschitz in z, Bernoulli, Volume 13 (2007) no. 1, pp. 80-91

[5] M.G. Crandall Viscosity solutions — a primer (I. Capuzzo Dolcetta; P.L. Lions, eds.), Viscosity Solutions and Applications, Lecture Notes in Mathematics, vol. 1660, Springer, Berlin, 1997, pp. 1-43

[6] S. He; J. Wang; J. Yan Semimartingal Theory and Stochastic Calculus, Science Press and CRC Press, Inc., 1992

[7] G. Jia A uniqueness theorem for the solution of backward stochastic differential equations, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008), pp. 439-444

[8] M. Kobylanski Backward stochastic differential equations and partial equations with quadratic growth, Ann. Probab., Volume 28 (2000), pp. 259-276

[9] J.P. Lepetier; J. San Martin Backward stochastic differential equations with continuous coefficient, Statistics and Probability Letters, Volume 32 (1997), pp. 425-430

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

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

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

[13] S. Peng Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer's type, Probability Theory and Related Fields, Volume 113 (1999), pp. 473-499

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Cité par 13 documents. Sources : Crossref, zbMATH

^☆ Supported by the National Natural Science Foundation of China (No. 10671205), National Basic Research Program of China (No. 2007CB814901), Youth Foundation of China University of Mining & Technology (Nos. 2006A041 and 2007A029) and Qing Lan Project.

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