Stochastic differential games with random coefficients and state constraints: associated stochastic Hamilton-Jacobi-Bellman-Isaacs equations with Neumann boundary conditions

Abstract

We study a stochastic differential game in which the state process is governed by a reflected stochastic differential equation (RSDE) and the recursive payoff is defined by a generalized backward stochastic differential equation (GBSDE). Unlike the Markovian models in \cite{Li08sicon,Fan}, all coefficients may depend directly on $ω$, so the value functions are random fields. We show that the lower and upper value functions are viscosity solutions of stochastic Hamilton--Jacobi--Bellman--Isaacs (HJBI) equations with nonlinear Neumann boundary conditions, thereby extending the framework of \cite{Qiu22AAP} to reflected state constraints. Under weak boundary regularity, we derive estimates for RSDEs and GBSDEs, prove regularity of the value functions, and establish the dynamic programming principle (DPP) as well as the existence of viscosity solutions. Finally, under additional assumptions in a one-dimensional linear setting, we prove a comparison theorem and the uniqueness of bounded $(L_y,0)$-viscosity solutions, where $L_y$ is the constant coefficient of $y$ in the running generator. To prove the comparison theorem, we homogenize the boundary condition and construct smooth adapted comparison fields by solving deterministic uniformly parabolic Neumann problems in finite dimensions.