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
[Unicité des solutions d'équations différentielles stochastiques multidimensionnelle ayant des générateurs uniformément continus]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2010-01-28
Accepté le : 2010-03-18
Publié le : 2010-05-23

DOI : 10.1016/j.crma.2010.03.013

Affiliations des auteurs :

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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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/

Bibliographie
Cité par

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[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

Shengjun Fan; Ying Hu; Shanjian Tang Existence, uniqueness and comparison theorem on unbounded solutions of scalar super-linear BSDEs, Stochastic Processes and their Applications, Volume 157 (2023), pp. 335-375 | DOI:10.1016/j.spa.2022.12.008 | Zbl:1520.60030
Yajie Chen; Chuanzhi Xing; Xiao Zhang $L^{p}$ solution of general mean-field BSDEs with continuous coefficients, Acta Mathematica Scientia. Series B. (English Edition), Volume 40 (2020) no. 4, pp. 1116-1140 | DOI:10.1007/s10473-020-0417-x | Zbl:1499.60180
Xiao Lishun; Fan Shengjun $L^{p} (p \geq 1)$ solutions of multidimensional BSDES with time-varying quasi-Hölder continuity generators in general time intervals, Communications of the Korean Mathematical Society, Volume 35 (2020) no. 2, pp. 667-684 | DOI:10.4134/ckms.c190272 | Zbl:1454.60082
Shengjun Fan $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions, Electronic Journal of Probability, Volume 24 (2019), p. 48 (Id/No 88) | DOI:10.1214/19-ejp345 | Zbl:1466.60117
Huan-Huan Luo; Sheng-Jun Fan Bounded solutions for general time interval BSDEs with quadratic growth coefficients and stochastic conditions, Stochastics and Dynamics, Volume 18 (2018) no. 5, p. 24 (Id/No 1850034) | DOI:10.1142/s021949371850034x | Zbl:1401.60115
Jiaojiao Ma; Shengjun Fan; Rui Fang On the stability theorem of $L^{p}$ solutions for multidimensional BSDEs with uniform continuity generators in $z$ , Brazilian Journal of Probability and Statistics, Volume 30 (2016) no. 2, pp. 321-344 | DOI:10.1214/15-bjps282 | Zbl:1341.60056
ShengJun Fan Bounded solutions, $L^{p}$ $(p > 1)$ solutions and $L^{1}$ solutions for one dimensional BSDEs under general assumptions, Stochastic Processes and their Applications, Volume 126 (2016) no. 5, pp. 1511-1552 | DOI:10.1016/j.spa.2015.11.012 | Zbl:1335.60087
Shengjun Fan; Yanbin Wang; Lishun Xiao Multidimensional BSDEs with uniformly continuous generators and general time intervals, Bulletin of the Korean Mathematical Society, Volume 52 (2015) no. 2, pp. 483-504 | DOI:10.4134/bkms.2015.52.2.483 | Zbl:1310.60077
ShengJun Fan $L^{p}$ solutions of multidimensional BSDEs with weak monotonicity and general growth generators, Journal of Mathematical Analysis and Applications, Volume 432 (2015) no. 1, pp. 156-178 | DOI:10.1016/j.jmaa.2015.06.049 | Zbl:1328.35338
Li-Shun Xiao; Sheng-Jun Fan; Na Xu $L^{p}$ $(p \geq 1)$ solutions of multidimensional BSDEs with monotone generators in general time intervals, Stochastics and Dynamics, Volume 15 (2015) no. 1, p. 34 (Id/No 1550002) | DOI:10.1142/s0219493715500021 | Zbl:1306.60082
Min Li; Yufeng Shi WITHDRAWN: Multidimensional BSDEs with uniformly continuous coefficients: The general result, Statistics Probability Letters (2014) | DOI:10.1016/j.spl.2014.02.014
Sheng Jun Fan; Long Jiang Multidimensional BSDEs with weak monotonicity and general growth generators, Acta Mathematica Sinica. English Series, Volume 29 (2013) no. 10, pp. 1885-1906 | DOI:10.1007/s10114-013-2128-x | Zbl:1274.60179
Pengju Duan; Yong Ren BSDES ON FINITE AND INFINITE TIME HORIZON WITH DISCONTINUOUS COEFFICIENTS, Bulletin of the Korean Mathematical Society, Volume 50 (2013) no. 4, p. 1079 | DOI:10.4134/bkms.2013.50.4.1079
Shengjun Fan; Long Jiang $L^{p}$ solutions to multidimensional backward stochastic differential equations with uniformly continuous generators, Chinese Annals of Mathematics. Series A, Volume 34 (2013) no. 6, pp. 761-770 | Zbl:1313.60098

Cité par 14 documents. Sources : Crossref, zbMATH

^☆ 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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