April 2023 Stochastic differential games with random coefficients and stochastic Hamilton–Jacobi–Bellman–Isaacs equations
Jinniao Qiu, Jing Zhang
Author Affiliations +
Ann. Appl. Probab. 33(2): 889-930 (April 2023). DOI: 10.1214/22-AAP1831


In this paper, we study a class of zero-sum two-player stochastic differential games with the controlled stochastic differential equations and the payoff/cost functionals of recursive type. As opposed to the pioneering work by Fleming and Souganidis [Indiana Univ. Math. J. 38 (1989) 293–314] and the seminal work by Buckdahn and Li [SIAM J. Control Optim. 47 (2008) 444–475], the involved coefficients may be random, going beyond the Markovian framework and leading to the random upper and lower value functions. We first prove the dynamic programming principle for the game, and then under the standard Lipschitz continuity assumptions on the coefficients, the upper and lower value functions are shown to be the viscosity solutions of the upper and the lower fully nonlinear stochastic Hamilton–Jacobi–Bellman–Isaacs (HJBI) equations, respectively. A stability property of viscosity solutions is also proved. Under certain additional regularity assumptions on the diffusion coefficient, the uniqueness of the viscosity solution is addressed as well.

Funding Statement

The first author was partially supported by the National Science and Engineering Research Council of Canada and by the start-up funds from the University of Calgary.
The second author is partially supported by National Key R&D Program of China (2018YFA0703900), National Natural Science Foundation of China (12031009) and Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education.


The authors are very grateful to the editor and the anonymous referees for their very valuable remarks and comments which have made it possible to improve our paper. J. Qiu would also thank Professor Hongjie Dong from Brown University and Jianfeng Zhang from the University of Southern California for helpful communications about classical solutions of deterministic Hamilton–Jacobi–Bellman–Isaacs equations.


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Jinniao Qiu. Jing Zhang. "Stochastic differential games with random coefficients and stochastic Hamilton–Jacobi–Bellman–Isaacs equations." Ann. Appl. Probab. 33 (2) 889 - 930, April 2023. https://doi.org/10.1214/22-AAP1831


Received: 1 March 2020; Revised: 1 January 2022; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 1518.91012
MathSciNet: MR4564416
Digital Object Identifier: 10.1214/22-AAP1831

Primary: 35D40 , 49L12 , 49L20 , 49L25 , 60H15 , 93E20

Keywords: backward stochastic partial differential equation , stochastic differential game , Stochastic Hamilton–Jacobi–Bellman–Isaacs equation , viscosity solution

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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