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A New Algorithm for Computing the Stabilizing Solution of General Periodic Time-Varying Stochastic Game-Theoretic Riccati Differential Equations

Yiyuan Wang

TL;DR

The paper tackles computing stabilizing periodic solutions to a broad class of time-varying stochastic game-theoretic Riccati differential equations (SGTRDEs) with periodic coefficients, originating from zero-sum linear-quadratic stochastic differential games. It introduces a dual-layer matrix-valued function iteration that reformulates the problem into interconnected bilevel subproblems, solved via maximal periodic solutions of associated Riccati equations. Under a discriminant condition that combines stochastic stabilizability and detectability, the method ensures existence and uniqueness of stabilizing solutions and proves convergence of the iterative sequence to the true stabilizing solution of the SGTRDE. Numerical experiments across dimensions 1–20 validate the approach, demonstrating robust convergence and stability in periodic settings. The result is a unified numerical framework applicable to deterministic, stochastic $H_ olinebreak_ olinebreak ext{-}infty$ SGTRDEs and ZSLQSDG-type problems with periodic coefficients, with practical impact on computing stable feedback gains for infinite-horizon, time-varying game systems.

Abstract

We propose a new algorithm for a broad class of periodic time-varying Stochastic Game-Theoretic Riccati Differential Equations arising in Zero-Sum Linear-Quadratic Stochastic Differential Games. The algorithm is constructed via dual-layer matrix-valued functions iteration sequences, which reformulate the original problem into a set of interconnected bilevel subproblems. By sequentially computing the maximal periodic solutions to the Riccati differential equations associated with each subproblem, we derive the stabilizing periodic solutions for the original problem and rigorously prove the algorithm's convergence. Numerical experiments verifies algorithm effectiveness and stability. This study provides a unified numerical framework for solving a wider range of periodic time-varying Stochastic Game-Theoretic Riccati Differential Equations.

A New Algorithm for Computing the Stabilizing Solution of General Periodic Time-Varying Stochastic Game-Theoretic Riccati Differential Equations

TL;DR

The paper tackles computing stabilizing periodic solutions to a broad class of time-varying stochastic game-theoretic Riccati differential equations (SGTRDEs) with periodic coefficients, originating from zero-sum linear-quadratic stochastic differential games. It introduces a dual-layer matrix-valued function iteration that reformulates the problem into interconnected bilevel subproblems, solved via maximal periodic solutions of associated Riccati equations. Under a discriminant condition that combines stochastic stabilizability and detectability, the method ensures existence and uniqueness of stabilizing solutions and proves convergence of the iterative sequence to the true stabilizing solution of the SGTRDE. Numerical experiments across dimensions 1–20 validate the approach, demonstrating robust convergence and stability in periodic settings. The result is a unified numerical framework applicable to deterministic, stochastic SGTRDEs and ZSLQSDG-type problems with periodic coefficients, with practical impact on computing stable feedback gains for infinite-horizon, time-varying game systems.

Abstract

We propose a new algorithm for a broad class of periodic time-varying Stochastic Game-Theoretic Riccati Differential Equations arising in Zero-Sum Linear-Quadratic Stochastic Differential Games. The algorithm is constructed via dual-layer matrix-valued functions iteration sequences, which reformulate the original problem into a set of interconnected bilevel subproblems. By sequentially computing the maximal periodic solutions to the Riccati differential equations associated with each subproblem, we derive the stabilizing periodic solutions for the original problem and rigorously prove the algorithm's convergence. Numerical experiments verifies algorithm effectiveness and stability. This study provides a unified numerical framework for solving a wider range of periodic time-varying Stochastic Game-Theoretic Riccati Differential Equations.

Paper Structure

This paper contains 11 sections, 8 theorems, 73 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. A Class of Stochastic Game-Theoretic Riccati Differential Equations arising in Zero-Sum Linear-Quadratic Stochastic Differential Games
  5. Algorithm Design
  6. A class of Auxiliary Problems
  7. The Main Results
  8. Analysis of Structural Characteristics for SGTRDEs
  9. Discriminant condition for stabilizing solutions of Riccati differential equations with definite-sign quadratic term
  10. Convergence Analysis of Iterative Sequence
  11. Numerical Experiments

Key Result

Proposition 3.1

Let $(t,X),(t,X+Z) \in \mathrm{Dom}\,\mathcal{G}$, then where

Figures (1)

  • Figure 1: Inner iteration counts for the first 8 outer iterations, ordered by computation sequence. The plot illustrates the progressive increase in inner iterations as convergence to the stabilizing solution, along with reduced variability of iterations in high-dimensional systems.

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2: Dragan2013book
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 11 more