Stochastic differential games with random coefficients and stochastic Hamilton–Jacobi–Bellman–Isaacs equations

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