Solvability for indefinite mean‐field stochastic linear quadratic optimal control with random jumps and its applications

Summary In this article, we study the open‐loop and closed‐loop solvability for indefinite mean‐field stochastic linear quadratic (MF‐SLQ) optimal control problem and its application in finance, where the controlled stochastic system is driven by a Brownian motion and a Poisson random martingale measure and also disturbed by some stochastic processes. The intrinsic property of stochastic systems results in the inequivalence of those two solvabilities, which is different from deterministic case. Based on a well‐posedness result of the problem, it is shown that the uniform convexity of cost functional is sufficient for the open‐loop solvability of the problem. By a matrix minimum principle, the necessity condition of regular solvability for a decoupled Riccati equation is established, meanwhile, the closed‐loop solvability is turned out to be equivalent to the regular solvability of Riccati equations with some constraints on the solutions of disturbances equations, moreover, the optimal closed‐loop strategy is characterized by regular solutions of Riccati equations and adapted solutions of disturbances equations. And then a mean‐variance model is considered to solve optimal portfolio selection strategy problem of the insurance company with liability. Our study fills a gap in the field of solvability for mean‐field stochastic optimal control with random jumps and provides a different way for solving mean‐variance portfolio selection model.