Probability theory
Uniqueness of solution to scalar BSDEs with $Lexp\left({\mu }_{0}\sqrt{2log\left(1+L\right)}\right)$-integrable terminal values: an ${L}^{1}$-solution approach
Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1085-1095.

This paper deals with a class of scalar backward stochastic differential equations (BSDEs) with $Lexp\left({\mu }_{0}\sqrt{2log\left(1+L\right)}\right)$-integrable terminal values for a critical parameter ${\mu }_{0}>0$. We show that the solution of these BSDEs is closely connected to the ${L}^{1}$-solution of the BSDEs with integrable parameters. The key tool is the Girsanov theorem. This idea leads to a new approach to the uniqueness of solution and we obtain a new existence and uniqueness result under general assumptions.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.236
Classification: 60H10
Hun O 1; Mun-Chol Kim 1; Chol-Gyu Pak 1

1. Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea
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Hun O; Mun-Chol Kim; Chol-Gyu Pak. Uniqueness of solution to scalar BSDEs with $L\protect \qopname{}{o}{exp}\left(\mu _0\protect \sqrt{2\protect \qopname{}{o}{log}(1+L)}\right)$-integrable terminal values: an $L^1$-solution approach. Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1085-1095. doi : 10.5802/crmath.236. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.236/

[1] Philippe Briand; Bernard Delyon; Ying Hu; Etienne Pardoux; Lucreţiu Stoica ${L}^{p}$ solutions of backward stochastic differential equations, Stochastic Processes Appl., Volume 108 (2003) no. 1, pp. 109-129 | Article | MR 2008603 | Zbl 1075.65503

[2] Philippe Briand; Ying Hu Quadratic BSDEs with convex generators and unbounded terminal conditions, Probab. Theory Relat. Fields, Volume 141 (2008) no. 3-4, pp. 543-567 | Article | MR 2391164 | Zbl 1141.60037

[3] Rainer Buckdahn; Ying Hu; Shanjian Tang Uniqueness of solution to scalar BSDEs with $Lexp\left(\mu \sqrt{2log\left(1+L\right)}\right)$-integrable terminal values, Electron. Commun. Probab., Volume 23 (2018), 59, 8 pages | Zbl 1414.60040

[4] Samuel N. Cohen; Robert J. Elliott Existence, uniqueness and comparisons for BSDEs in general spaces, Ann. Probab., Volume 40 (2012) no. 5, pp. 2264-2297 | MR 3025717 | Zbl 1260.60128

[5] Shengjun Fan ${L}^{p}$ solutions of multidimensional BSDEs with weak monotonicity and general growth generators, J. Math. Anal. Appl., Volume 432 (2015) no. 1, pp. 156-178 | MR 3371229 | Zbl 1328.35338

[6] Shengjun Fan Bounded solutions, ${L}^{p}$ ($p>1$) solutions and ${L}^{1}$ solutions for one-dimensional BSDEs under general assumptions, Stochastic Processes Appl., Volume 126 (2016) no. 5, pp. 1511-1552 | MR 3473104 | Zbl 1335.60087

[7] Shengjun Fan Existence, Uniqueness and Stability of ${L}^{1}$ Solutions for Multidimensional Backward Stochastic Differential Equations with Generators of One-Sided Osgood Type, J. Theor. Probab., Volume 31 (2018) no. 3, pp. 1860-1899 | MR 3842172 | Zbl 1404.60080

[8] Shengjun Fan; Ying Hu Existence and uniqueness of solution to scalar BSDEs with $Lexp\left(\mu \sqrt{2log\left(1+L\right)}\right)$-integrable terminal values: the critical case, Electron. Commun. Probab., Volume 24 (2019), 49, 10 pages | Zbl 1422.60094

[9] Shengjun Fan; Long Jiang Multidimensional BSDEs with weak monotonicity and general growth generators, Acta Math. Sin., Engl. Ser., Volume 29 (2013) no. 10, pp. 1885-1906 | MR 3096551 | Zbl 1274.60179

[10] Y. Hu; Shanjian Tang Existence of solution to scalar BSDEs with $Lexp\left(\sqrt{\frac{2}{\lambda }log\left(1+L\right)}\right)$-integrable terminal values, Electron. Commun. Probab., Volume 23 (2018), 27 | Zbl 1390.60208

[11] Norihiko Kazamaki Continuous Exponential Martingales and BMO, Lecture Notes in Mathematics, 1579, Springer, 1994 | MR 1299529 | Zbl 0806.60033

[12] Etienne Pardoux; Shige Peng Adapted solution of a backward stochastic differential equation, Syst. Control Lett., Volume 14 (1990) no. 1, pp. 55-61 | Article | MR 1037747 | Zbl 0692.93064

[13] Etienne Pardoux; Aurel Răşcanu Stochastic differential equations, Backward SDEs, Partial differential equations, Stochastic Modelling and Applied Probability, 69, Springer, 2014 | MR 3308895 | Zbl 1321.60005

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