A Nested Iterative Algorithm for Zero-Sum Linear-Quadratic Stochastic Differential Games in an Infinite Horizon

This paper proposes a new algorithm to compute closed-loop saddle points for infinite-horizon zero-sum linear-quadratic stochastic differential games via structural decoupling. Specifically, we develop a nested iterative scheme that generates a monotonically increasing matrix sequence to decompose the original problem into coupled subproblems. By sequentially solving the stabilizing solutions of associated algebraic Riccati equations for each subproblem, we recover the original problem's stabilizing solution and rigorously prove sequence convergence. 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.