On the stochastic linear quadratic control problem with piecewise constant admissible controls

Abstract

A linear quadratic optimal control problem for a system described by Itô differential equations with state and control dependent white noise under the assumption that the set of admissible controls consists of a class of piecewise constant stochastic processes is considered. The considered LQ optimal control problem is converted into a LQ optimization problem for a stochastic controlled system with finite jumps and multiplicative white noise perturbations. One of the original contribution of this work is the proof of the equivalence between the solvability of the considered optimal control problem and the solvability of the problem with given terminal values associated to a matrix linear differential equation (MLDE) with finite jumps and constraints. Another original contribution consists in the proof of the global existence of the solution of the problem with given terminal value of the MLDE if the cost weights matrices are positive semidefinite. The results obtained in the case of a LQ optimal control for systems with finite jumps are then applied to derive explicit formulae of the optimal controls for the optimization problem under piecewise constant controls.