A Large-Population Stochastic Differential Game With Terminal State Constraint and Common Noise

Abstract

In this article, we focus on a type of linear–quadratic (LQ) mean-field game of stochastic differential equations with a terminal state constraint and common noise, where a coupling structure enters the state equation, cost functional, and constraint condition. First, by virtue of the mean-field method, we introduce an auxiliary problem of the original game, which is a constrained optimal control problem. Second, by virtue of the Lagrangian multiplier method and stochastic maximum principle, a decentralized control strategy depending on the optimal Lagrangian multiplier is derived. Finally, we prove that the decentralized control strategy obtained is an $\epsilon$-Nash equilibrium of the LQ mean-field game. As an application, we solve a financial problem and give some numerical results.