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Linear‐Quadratic Optimal Control Problems for Mean‐Field Stochastic Differential Equations with Jumps
Author(s) -
Tang Maoning,
Meng Qingxin
Publication year - 2019
Publication title -
asian journal of control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.769
H-Index - 53
eISSN - 1934-6093
pISSN - 1561-8625
DOI - 10.1002/asjc.1762
Subject(s) - mathematics , stochastic differential equation , stochastic control , martingale (probability theory) , optimal control , stochastic partial differential equation , uniqueness , brownian motion , linear quadratic gaussian control , mathematical analysis , differential equation , mathematical optimization , statistics
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 one‐dimensional 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 equations and the stochastic Hamilton system and then prove the optimal control has a state feedback representation.