Non‐zero‐sum differential games of delayed backward doubly stochastic systems and their application

Abstract

AbstractIn this paper, we investigate the non‐zero‐sum differential game problem based on the delayed backward doubly stochastic differential equation. We discuss the case where the controlled system contains time‐delayed variables, and the time‐delayed variables are different. We construct a new kind of adjoint equation consisting of a doubly stochastic differential equation and three simple differential equations. Under appropriate premises for such equations, we deduce a stochastic maximum principle as a necessary condition and a verification theorem for the Nash equilibrium point by using the duality method and convex variation technique. As an application, we apply our result to a linear delayed backward doubly stochastic non‐zero‐sum game problem to verify the effectiveness of our results.