Stochastic linear quadratic optimal control with partial information and its application to mean-variance hedging problems

Abstract

This paper is concerned with a stochastic linear quadratic (LQ) optimal control with partial information where the control system is a non-Markov process. We solved this problem explicitly by completion of squares method. An optimal control is denoted by the corresponding optimal state equation, a Riccati differential equation and a backward stochastic differential equation (BSDE) with the dynamics similar to the optimal state equation. And then the general result is applied to a partial information mean-variance hedging problem, where an optimal mean-variance portfolio strategy is denoted by the sum of a replicating portfolio strategy for a contingent claim and a Merton’s portfolio strategy with partial information. By filtering for SDEs, an explicitly observable optimal portfolio strategy for a partial information mean-variance hedging problem is presented, and some numerical simulations on the problem are given to furthermore support our theoretical results.