Stabilization and Optimality of Pursuer and Evader Policies for Stochastic Differential Games

This paper examines a two‐player, zero‐sum stochastic differential game problem over an infinite time horizon, where the players employ controller (pursuer) and stopper (evader) policies for a nonlinear system driven by Brownian motion. The pursuer's goal is to minimize a nonlinear‐nonquadratic performance criterion while guaranteeing stochastic stabilization, and the evader's goal is to maximize the performance criterion. By establishing a connection between stochastic Lyapunov stability theory and the stochastic Hamilton–Jacobi–Isaacs equation, we derive explicit optimal strategies for both players that guarantee stochastic stabilization as well as enforcing a saddle‐point condition on a nonlinear‐nonquadratic performance criterion. We demonstrate that global asymptotic stability in probability is ensured through a Lyapunov function, which also serves as the solution to the steady‐state stochastic Hamilton–Jacobi–Isaacs equation, thereby guaranteeing both closed‐loop stability in probability and optimality. Furthermore, we develop optimal feedback controllers and stopper policies for affine nonlinear systems using an inverse optimality framework tailored to the game problem. These results extend existing linear feedback pursuer‐evader policies to nonlinear controllers and stoppers that optimize general polynomial and multilinear performance criteria.