Multiscale Linear-Quadratic Stochastic Optimal Control with multiplicative noise

Beniamin Goldys, Gianmario Tessitore, James Yang and Zhou Zhou

Abstract

We investigate asymptotic properties of a finite-time horizon stochastic control when the cost functional is quadratic and the state process is given by a two component slow-fast system that is linear in drift and and in the Brownian diffusion term. We approach this problem by considering the associated differential Riccati equation and reformulating it as a deterministic singular perturbation problem. Asymptotic properties of this deterministic problem are deduced from a version of the Tikhonov Theorem. We propose two methods to approximate the value function of the stochastic optimal control problem. The first one is by constructing an approximately optimal control process whilst the second one is by identifying the limit to the value function. Both approximation methods rely on a new result on the existence of a solution to a coupled differential-algebraic Riccati equation with certain stability properties.

Keywords: singular perturbations, linear-quadratic stochastic optimal control, differential-algebraic Riccati equation, slow-fast system multiscale system.

AMS Subject Classification: Primary 34E13; secondary 34D15, 93C70, 93E20.