Comptes Rendus
no. 2
Probabilités
On the uniqueness of solutions to quadratic BSDEs with non-convex generators and unbounded terminal conditions
Shengjun Fan1 ; Ying Hu2 ; Shanjian Tang3
1 School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
2 Univ. Rennes, CNRS, IRMAR-UMR6625, F-35000, Rennes, France
3 Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 227-235.

We prove a uniqueness result of the unbounded solution for a quadratic backward stochastic differential equation whose terminal condition is unbounded and whose generator g may be non-Lipschitz continuous in the state variable y and non-convex (non-concave) in the state variable z, and instead satisfies a strictly quadratic condition and an additional assumption. The key observation is that if the generator is strictly quadratic, then the quadratic variation of the first component of the solution admits an exponential moment. Typically, a Lipschitz perturbation of some convex (concave) function satisfies the additional assumption mentioned above. This generalizes some results obtained in [1] and [2].

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DOI : 10.5802/crmath.40
Classification : 60H10

Shengjun Fan 1 ; Ying Hu 2 ; Shanjian Tang 3

1 School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
2 Univ. Rennes, CNRS, IRMAR-UMR6625, F-35000, Rennes, France
3 Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Shengjun Fan; Ying Hu; Shanjian Tang. On the uniqueness of solutions to quadratic BSDEs with non-convex generators and unbounded terminal conditions. Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 227-235. doi : 10.5802/crmath.40. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.40/

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  • Chuang Gu; Yan Wang; Shengjun Fan Existence, uniqueness and comparison theorem on unbounded solutions of general time-interval BSDEs with sub-quadratic generators, Probability, Uncertainty and Quantitative Risk, Volume 10 (2025) no. 1, p. 31 | DOI:10.3934/puqr.2025003
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  • Shengjun Fan; Ying Hu; Shanjian Tang Multi-dimensional backward stochastic differential equations of diagonally quadratic generators: The general result, Journal of Differential Equations, Volume 368 (2023), p. 105 | DOI:10.1016/j.jde.2023.05.041
  • 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), p. 335 | DOI:10.1016/j.spa.2022.12.008
  • Camilo Hernández On quadratic multidimensional type-I BSVIEs, infinite families of BSDEs and their applications, Stochastic Processes and their Applications, Volume 162 (2023), p. 249 | DOI:10.1016/j.spa.2023.05.001
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  • Ying Hu; Remi Moreau; Falei Wang Quadratic mean-field reflected BSDEs, Probability, Uncertainty and Quantitative Risk, Volume 7 (2022) no. 3, p. 169 | DOI:10.3934/puqr.2022012
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