×

zbMATH — the first resource for mathematics

Existence and uniqueness of solution to scalar BSDEs with \(L\exp (\mu \sqrt{2\log (1+L)} )\)-integrable terminal values: the critical case. (English) Zbl 1422.60094
Summary: In [Y. Hu and S. Tang, Electron. Commun. Probab. 23, Paper No. 27, 11 p. (2018; Zbl 1390.60208)], the existence of the solution is proved for a scalar linearly growingbackward stochastic differential equation (BSDE) when the terminal value is\(L\exp (\mu \sqrt{2\log (1+L)} )\)-integrable for a positive parameter \(\mu >\mu _0\) with a critical value \(\mu _0\), and a counterexample is provided to show that the preceding integrability for \(\mu <\mu _0\) is not sufficient to guarantee the existence of the solution. Afterwards, the uniqueness result (with \(\mu >\mu _0)\) is also given in [R. Buckdahn et al., Electron. Commun. Probab. 23, Paper No. 59, 8 p. (2018; Zbl 1414.60040)] for the preceding BSDE under the uniformly Lipschitz condition of the generator. In this note, we prove that these two results still hold for the critical case: \(\mu =\mu _0\).
MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Philippe Briand, Bernard Delyon, Ying Hu, Etienne Pardoux, and L. Stoica, \(L^p\) solutions of backward stochastic differential equations, Stochastic Process. Appl. 108 (2003), no. 1, 109-129. · Zbl 1075.65503
[2] Philippe Briand and Ying Hu, BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields 136 (2006), no. 4, 604-618. · Zbl 1109.60052
[3] Rainer Buckdahn, Ying Hu, and Shanjian Tang, Existence of solution to scalar BSDEs with \({L}\exp \left (\mu \sqrt{2\log (1+L)} \right )\)-integrable terminal values, Electron. Commun. Probab. 23 (2018), Paper No. 59, 8pp. · Zbl 1414.60040
[4] Freddy Delbaen, Ying Hu, and Adrien Richou, On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions, Ann. Inst. Henri PoincarĂ© Probab. Stat. 47 (2011), 559-574. · Zbl 1225.60093
[5] Nicole El Karoui, Shige Peng, and Marie Claire Quenez, Backward stochastic differential equations in finance, Math. Finance 7 (1997), no. 1, 1-71. · Zbl 0884.90035
[6] Shengjun Fan, Bounded solutions, \({L}^p\ (p>1)\) solutions and \({L}^1\) solutions for one-dimensional BSDEs under general assumptions, Stochastic Process. Appl. 126 (2016), 1511-1552. · Zbl 1335.60087
[7] Shengjun Fan and Long Jiang, \({L}^p (p>1)\) solutions for one-dimensional BSDEs with linear-growth generators, Journal of Applied Mathematics and Computing 38 (2012), no. 1-2, 295-304. · Zbl 1296.60147
[8] Ying Hu and Shanjian Tang, Existence of solution to scalar BSDEs with \({L}\exp \sqrt{{2\over \lambda }\log (1+L)} \)-integrable terminal values, Electron. Commun. Probab. 23 (2018), Paper No. 27, 11pp. · Zbl 1414.60040
[9] Jean-Pierre Lepeltier and Jaime San Martin, Backward stochastic differential equations with continuous coefficient, Statist. Probab. Lett. 32 (1997), no. 4, 425-430. · Zbl 0904.60042
[10] Etienne Pardoux and Shige Peng, Adapted solution of a backward stochastic differential equation, Syst. Control Lett. 14 (1990), no. 1, 55-61. · Zbl 0692.93064
[11] Shanjian Tang, Dual representation as stochastic differential games of backward stochastic differential equations and dynamic evaluations, C. R. Math. Acad. Sci. Paris 342 (2006), 773-778. · Zbl 1130.91015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.