Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions

Freddy Delbaen, Ying Hu, and Adrien Richou

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Abstract

In [Probab. Theory Related Fields 141 (2008) 543–567], the authors proved the uniqueness among the solutions of quadratic BSDEs with convex generators and unbounded terminal conditions which admit every exponential moments. In this paper, we prove that uniqueness holds among solutions which admit some given exponential moments. These exponential moments are natural as they are given by the existence theorem. Thanks to this uniqueness result we can strengthen the nonlinear Feynman–Kac formula proved in [Probab. Theory Related Fields 141 (2008) 543–567].

Résumé

Les auteurs de l’article [Probab. Theory Related Fields 141 (2008) 543–567] ont prouvé un résultat d’unicité pour les solutions d’EDSRs quadratiques de générateur convexe et de condition terminale non bornée ayant tous leurs moments exponentiels finis. Dans ce papier, nous prouvons que ce résultat d’unicité reste vrai pour des solutions qui admettent uniquement certains moments exponentiels finis. Ces moments exponentiels sont reliés de manière naturelle à ceux présents dans le théorème d’existence. À l’aide de ce résultat d’unicité nous pouvons améliorer la formule de Feynman–Kac non linéaire prouvée dans [Probab. Theory Related Fields 141 (2008) 543–567].

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 2 (2011), 559-574.

Dates
First available in Project Euclid: 23 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1300887282

Digital Object Identifier
doi:10.1214/10-AIHP372

Mathematical Reviews number (MathSciNet)
MR2814423

Zentralblatt MATH identifier
1225.60093

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Backward stochastic differential equations Generator of quadratic growth Unbounded terminal condition Uniqueness result Nonlinear Feynman–Kac formula

Citation

Delbaen, Freddy; Hu, Ying; Richou, Adrien. On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 2, 559--574. doi:10.1214/10-AIHP372. https://projecteuclid.org/euclid.aihp/1300887282


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