标题： 带有跳跃的均值场随机微分方程的线性二次最优控制问题 显示英文标题

标题： Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations with Jumps

摘要： 在本文中，我们研究由泊松随机鞅测度和多维布朗运动驱动的均值场随机微分方程的线性二次最优控制问题。 首先，通过经典的凸变分原理得到了最优控制的存在性和唯一性。 其次，通过对偶方法，推导出最优系统，也称为随机哈密顿系统，该系统被证明是一个带有跳跃的线性完全耦合的均值场前向后向随机微分方程，用于表征最优控制。 第三，应用解耦技术，我们建立了两个黎卡提方程与随机哈密顿系统之间的联系，然后证明最优控制具有状态反馈表示形式。 显示英文摘要

摘要： In this paper, we study a linear-quadratic optimal control problem for mean-field stochastic differential equations driven by a Poisson random martingale measure and a multidimensional Brownian motion. Firstly, the existence and uniqueness of the optimal control is obtained by the classic convex variation principle. Secondly, by the duality method, the optimality system, also called the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps, is derived to characterize the optimal control. Thirdly, applying a decoupling technique, we establish the connection between two Riccati equation and the stochastic Hamilton system and then prove the optimal control has a state feedback representation.