Singular perturbations of zero-sum linear-quadratic stochastic differential games

Beniamin Goldys, James Yang and Zhou Zhou

Abstract

We investigate a class of zero-sum linear-quadratic stochastic differential games on a finite time horizon governed by multiscale state equations that describe the behaviour of fast and slow players. The multiscale nature of the problem can be leveraged to reformulate the associated generalised Riccati equation as a deterministic singular perturbation problem. We show that, for the fast time scale fast enough, the existence of solution to the associated generalised Riccati equation is guaranteed by the existence of a solution to a decoupled pair of differential and algebraic Riccati equations with a reduced order of dimensionality. Furthermore, we identify a pair of asymptotic estimates to the value function of the game problem by constructing an approximate feedback strategy and observing the limiting value function.

Keywords: singular perturbation, stochastic differential game, Riccati equation, linear-quadratic problem, zero-sum game, slow-fast, multiscale.