The open‐loop and closed‐loop Nash equilibrium of the local and remote stochastic game for multiplicative noise systems with inconsistent information

Abstract

Abstract In this article, the local and remote stochastic nonzero‐sum game for a multiplicative noise system with inconsistent information is investigated, in which the multiplicative noise can cause nonlinear characteristics of linear systems, making it difficult to solve the optimal linear feedback Nash equilibrium. For the considered local and remote stochastic nonzero‐sum game, the local player and the remote player obtain different information sets, which leads to inconsistent information between the two players. The goal is that each player is desired to minimize their own cost function. Our approach is based on a combination of orthogonal decomposition and completing square techniques, which allow us to derive a set of coupled Riccati equations that characterize the optimal feedback explicit (closed‐loop) Nash equilibrium. The contributions of this article are summarized as follows. First, the optimal open‐loop Nash equilibrium is obtained in terms of the forward and backward stochastic difference equations (FBSDEs) by adopting the Pontryagin maximum principle. Second, the closed‐loop Nash equilibrium of this local and remote stochastic nonzero‐sum game for a multiplicative noise system with inconsistent information is obtained by using the orthogonal decomposition methods. Finally, a simulation example is given to illustrate the validity of theoretical results and discuss potential extensions to more complex systems.