Indefinite Linear-Quadratic Optimal Control of Mean-Field Stochastic Differential Equation With Jump Diffusion: An Equivalent Cost Functional Method

In this article, we consider a linear-quadratic optimal control problem of mean-field stochastic differential equation with jump diffusion, which is also called as an mean-field linear-quadratic problem with jump diffusion (MF-LQJ) problem. Here, cost functional is allowed to be indefinite. We use an equivalent cost functional method to deal with the MF-LQJ problem with indefinite weighting matrices. Some equivalent cost functionals enable us to establish a bridge between indefinite and positive-definite MF-LQJ problems. With such a bridge, solvabilities of stochastic Hamiltonian system and Riccati equations are further characterized. Optimal control of the indefinite MF-LQJ problem is represented as a state feedback via solutions of Riccati equations. As a by-product, the method provides a new way to prove the existence and uniqueness of solution to mean-field forward-backward stochastic differential equation with jump diffusion, where existing methods in literatures do not work. Some examples are provided to illustrate our results.