# Stochastic differential games with state constraints and Isaacs equations with nonlinear Neumann problems

## TL;DR

This paper studies a two-player stochastic differential game with state constraints, establishing the value function as the unique viscosity solution of nonlinear Neumann boundary Isaacs equations, using a novel approach involving GBSDE generator representation.

## Contribution

It introduces a new method for proving viscosity solutions of Isaacs equations with nonlinear Neumann boundary conditions via GBSDE generator representation.

## Key findings

- Proves the dynamic programming principle for constrained stochastic games.
- Establishes the value function as the unique viscosity solution of the Isaacs equation.
- Develops a novel approach using random time change for GBSDE generator representation.

## Abstract

We investigate a two-player zero-sum stochastic differential game problem with the state process being constrained in a connected bounded closed domain, and the cost functional described by the solution of a generalized backward stochastic differential equation (GBSDE for short). We show that the value functions enjoy a (strong) dynamic programming principle, and are the unique viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs equations with nonlinear Neumann boundary problems. To obtain the existence for viscosity solutions, we provide a new approach utilizing the representation theorem for generators of the GBSDE, which is proved by a random time change method and is a novel result in its own right.