Singular Perturbation of Zero-Sum Linear-Quadratic Stochastic Differential Games

TL;DR

This paper studies zero-sum linear-quadratic stochastic differential games with multiscale dynamics, showing how to simplify the Riccati equations via singular perturbation techniques to analyze the game's value and strategies.

Contribution

It introduces a novel approach to handle multiscale stochastic differential games by reformulating Riccati equations as singular perturbation problems, enabling reduced-order analysis.

Findings

Existence of solutions guaranteed for small epsilon via decoupled Riccati equations.

Asymptotic estimates for the value function are derived.

Approximate feedback strategies are constructed based on the analysis.

Abstract

We investigate a class of zero-sum linear-quadratic stochastic differential games on a finite time horizon governed by multiscale state equations. The multiscale nature of the problem can be leveraged to reformulate the associated generalised Riccati equation as a deterministic singular perturbation problem. In doing so, we show that, for small enough $\epsilon$, 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 are able to formulate 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.