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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.

This note is dedicated to a kind of partially observable linear–quadratic control problem with model uncertainty, where the coefficients of cost functional are uncertain representing different market conditions. By virtue of backward separation technique, stochastic maximum principle, as well as filtering method, a feedback form of candidate optimal control is designed. Moreover, through some delicate analysis, the existence of maximal reference probability $\bar{\lambda }^*$ is certified. A considerable innovation of this note is the characterization of $\bar{\lambda }^*$ by computation of the optimal cost. Finally, a numerical simulation is presented to authenticate our theoretical results.