N -Player Stochastic Differential Games with Regime Switching and Mean Field Convergence
研究了机制转换下N人随机微分博弈的纳什均衡存在唯一性，并证明平均场博弈的最优控制可作为N人问题的近似纳什均衡。
In this study, we investigate N-player stochastic differential games with regime switching, in which the player dynamics are modulated by a finite-state Markov chain. We analyze the associated Nash system, which consists of a system of coupled nonlinear partial differential equations, and establish the existence and uniqueness of solutions to this system, thereby proving the existence of a unique Nash equilibrium. Additionally, we examine the mean field game (MFG) problem under the same regime-switching framework. We derive a connection between the Nash equilibrium of the MFG and a forward-backward stochastic differential equation with jumps and demonstrate the unique solvability of this equation. Finally, we explore the propagation of chaos and show that the optimal control obtained from the MFG serves as an approximate Nash equilibrium for the N-player problem. Funding: P. Chakraborty acknowledges support from the National Science Foundation [Grant DMS-2153915] and the Korb Early Career Professorship.