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

Shengjun Fan; Ying Hu; Shanjian Tang

doi:10.5802/crmath.40

Probability Theory

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

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

Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 227-235.

Abstract

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].

Received: 2019-05-28
Revised: 2020-03-17
Accepted: 2020-03-17
Published online: 2020-06-15

DOI: 10.5802/crmath.40

Classification: 60H10

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2020__358_2_227_0,
     author = {Shengjun Fan and Ying Hu and Shanjian Tang},
     title = {On the uniqueness of solutions to quadratic {BSDEs} with non-convex generators and unbounded terminal conditions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {227--235},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {2},
     year = {2020},
     doi = {10.5802/crmath.40},
     language = {en},
}

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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/

References
Cited by

[1] Philippe Briand; Ying Hu BSDE with quadratic growth and unbounded terminal value, Probab. Theory Relat. Fields, Volume 136 (2006) no. 4, pp. 604-618 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[3] Philippe Briand; Adrien Richou On the uniqueness of solutions to quadratic BSDEs with non-convex generators (2017) (https://arxiv.org/abs/1801.00157v1)

[4] Freddy Delbaen; Ying Hu; Adrien Richou On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions, Ann. Inst. Henri Poincaré, Volume 47 (2011) no. 2, pp. 559-574 | DOI | Numdam | MR | Zbl

[5] Freddy Delbaen; Ying Hu; Adrien Richou On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions: the critical case, Discrete Contin. Dyn. Syst., Volume 35 (2015) no. 11, pp. 5273-5283 | DOI | MR | Zbl

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[8] Adrien Richou Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition, Stochastic Processes Appl., Volume 122 (2012) no. 9, pp. 3173-3208 | DOI | MR | Zbl

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