A Unified Computational Approach for Zero-Sum Linear-Quadratic Stochastic Differential Games in Infinite Horizons

TL;DR

This paper introduces a novel iterative method for solving zero-sum linear-quadratic stochastic differential games in infinite horizons, enabling the computation of saddle points through Riccati equations.

Contribution

It develops a unified, convergent computational scheme that decomposes complex stochastic differential games into manageable subproblems, extending classical approaches.

Findings

The proposed method converges to the stabilizing solution.

Numerical example demonstrates effectiveness.

First general-purpose computational approach for this problem class.

Abstract

This paper proposes a new method for finding closed-loop saddle points in zero-sum linear-quadratic stochastic differential games by decoupling their inherent structure. Specifically, we develop a nested iterative scheme that constructs a monotonically increasing sequence of matrices, thereby decomposing the original problem into interconnected subproblems. By sequentially computing the stabilizing solutions to the algebraic Riccati equations within each subproblem, we obtain the stabilizing solution to the original problem and rigorously establish the convergence of the iterative sequence. A numerical example further validates the effectiveness of the proposed method. To the best of our knowledge, this work extends the classical setting and provides the first general-purpose computational approach for this class of problems.