Article Dans Une Revue The Annals of Applied Probability Année : 2025

Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth

Tao Hao
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Ying Hu
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  • IdHAL : ying-hu
Shanjian Tang
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  • PersonId : 968042
Jiaqiang Wen
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Résumé

In this paper, we study general mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. First, the existence and uniqueness of local and global solutions are proved with some new ideas for a one-dimensional mean-field BSDE when the generator $g\big(t, Y, Z, \mathbb{P}_{Y}, \mathbb{P}_{Z}\big)$ has a quadratic growth in $Z$ and the terminal value is bounded. Second, a comparison theorem for the general mean-field BSDEs is obtained with the Girsanov transform. Third, we prove the convergence of the particle systems to the mean-field BSDEs with quadratic growth, and the convergence rate is also given. Finally, in this framework, we use the mean-field BSDE to provide a probabilistic representation for the viscosity solution of a nonlocal partial differential equation (PDE, for short) as an extended nonlinear Feynman-Kac formula, which yields the existence and uniqueness of the solution to the PDE.

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hal-04966579 , version 1 (25-02-2025)

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Tao Hao, Ying Hu, Shanjian Tang, Jiaqiang Wen. Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth. The Annals of Applied Probability, 2025, 35 (3), pp.2128-2174. ⟨10.1214/25-AAP2167⟩. ⟨hal-04966579⟩
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