一类平均场型随机微分方程的存在唯一性定理

Existence and Uniqueness Theorem for a Class of Mean-Field Stochastic Differential Equations

本文研究一类由布朗运动驱动的平均场型随机微分方程,此方程与平均场型最优控制问题紧密相关,应用压缩映射定理给出了平均场型方程的存在唯一性定理,即当系统的系数关于状态和平均场满足Lipschitz连续,关于时间平方可积,并且初始状态变量满足一定可积性条件时,方程具有唯一的强解。

This paper explores a class of mean-field stochastic differential equations (SDEs) driven by Brownian motion, which is closely linked with mean-field optimal stochastic control problems. We utilize the contraction mapping theorem to establish the existence and uniqueness of solutions for the mean-field SDEs. That is, when the coefficients of the system with respect to the state and the mean-field satisfy Lipschitz continuity, are square integrable with respect to time, and the initial state variables satisfy certain integrability conditions, the equation has a unique strong solution.