A mean-field stochastic linear-quadratic optimal control problem with jumps under partial information

Abstract

In this article, the stochastic linear-quadratic optimal control problem of mean-field type with jumps under partial information is discussed. The state equation which contains affine terms is a SDE with jumps driven by a multidimensional Brownian motion and a Poisson stochastic martingale measure, and the quadratic cost function contains cross terms. In addition, the state and the control as well as their expectations are contained both in the state equation and the cost functional. This is the so-called optimal control problem of mean-field type. Firstly, the existence and uniqueness of the optimal control is proved. Secondly, the adjoint processes of the state equation is introduced, and by using the duality technique, the optimal control is characterized by the stochastic Hamiltonian system. Thirdly, by applying a decoupling technology, we deduce two integro-differential Riccati equations and get the feedback representation of the optimal control under partial information. Fourthly, the existence and uniqueness of the solutions of two Riccati equations are proved. Finally, we discuss a special case, and establish the corresponding feedback representation of the optimal control by means of filtering technique.