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A Convergent Algorithm Based on Deterministic Approximation for a Large Class of Regime-Switching Generalized Stochastic Game-Theoretic Riccati Differential Equations

Yiyuan Wang

TL;DR

The paper tackles the challenge of computing the stabilizing solution for regime-switching stochastic GTRDEs arising in infinite-horizon zero-sum LQ differential games with multiplicative noise and periodic coefficients. It introduces a convergent algorithm that decomposes the stochastic problem into deterministic subproblems, constructing a sequence of minimal PSD Riccati solutions via updates $M^{(h)}=oldsymbol{ extPi}_1igl[oldsymbol{X}^{(h-1)}igr]+M$, $L^{(h)}=oldsymbol{ extPi}_2igl[oldsymbol{X}^{(h-1)}igr]+L$, and $R^{(h)}=oldsymbol{ extPi}_3igl[oldsymbol{X}^{(h-1)}igr]+R$. The authors prove monotonicity, boundedness, and convergence of the iterates to the unique θ-periodic stabilizing solution, and they validate the method with numerical experiments demonstrating robustness across regimes and system sizes. The work unifies and extends classical ZSLQSDG settings, $H_ ext{∞}$-type stochastic GTRDEs with regime-switching, and broader GTRDEs in regime-switching contexts, providing a practical computational framework for adversarial control under uncertainty. This approach offers a general, provably convergent tool for obtaining stabilizing feedback in complex, regime-dependent stochastic games with multiplicative noise, with potential for wide application in finance and engineering systems subject to regime shifts.

Abstract

This paper proposes a novel iterative algorithm to compute the stabilizing solution of regime-switching stochastic game-theoretic Riccati differential equations with periodic coefficients. The method decomposes the original complex stochastic problem into a sequence of deterministic subproblems. By sequentially solving for the minimal solutions of the Riccati differential equations in each subproblem, a sequence of matrix-valued functions is constructed. Leveraging the comparison theorem, the monotonicity, boundedness, and convergence of the iterative sequence are rigorously proven. Numerical experiments verifies algorithm effectiveness and stability. To the best of our knowledge, this is the first general computational approach developed for this class of problems.

A Convergent Algorithm Based on Deterministic Approximation for a Large Class of Regime-Switching Generalized Stochastic Game-Theoretic Riccati Differential Equations

TL;DR

The paper tackles the challenge of computing the stabilizing solution for regime-switching stochastic GTRDEs arising in infinite-horizon zero-sum LQ differential games with multiplicative noise and periodic coefficients. It introduces a convergent algorithm that decomposes the stochastic problem into deterministic subproblems, constructing a sequence of minimal PSD Riccati solutions via updates , , and . The authors prove monotonicity, boundedness, and convergence of the iterates to the unique θ-periodic stabilizing solution, and they validate the method with numerical experiments demonstrating robustness across regimes and system sizes. The work unifies and extends classical ZSLQSDG settings, -type stochastic GTRDEs with regime-switching, and broader GTRDEs in regime-switching contexts, providing a practical computational framework for adversarial control under uncertainty. This approach offers a general, provably convergent tool for obtaining stabilizing feedback in complex, regime-dependent stochastic games with multiplicative noise, with potential for wide application in finance and engineering systems subject to regime shifts.

Abstract

This paper proposes a novel iterative algorithm to compute the stabilizing solution of regime-switching stochastic game-theoretic Riccati differential equations with periodic coefficients. The method decomposes the original complex stochastic problem into a sequence of deterministic subproblems. By sequentially solving for the minimal solutions of the Riccati differential equations in each subproblem, a sequence of matrix-valued functions is constructed. Leveraging the comparison theorem, the monotonicity, boundedness, and convergence of the iterative sequence are rigorously proven. Numerical experiments verifies algorithm effectiveness and stability. To the best of our knowledge, this is the first general computational approach developed for this class of problems.

Paper Structure

This paper contains 9 sections, 3 theorems, 54 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notation
  4. A Class of Regime-Switching Stochastic Game-Theoretic Riccati Differential Equations
  5. Some Useful Preliminary Results
  6. The Main Results
  7. Algorithm Design
  8. Convergence of the Iterative Sequence
  9. Numerical Experiments

Key Result

Theorem 2.4

Let $\mathbb{X}^2(\cdot) : I_2 \subseteq \mathbb{R}_+ \to S^N_n$ be a solution of the Stochastic GTRDEs pf:gtrde associated to the quintuple $\Sigma^2 = (\mathbb{\hat{A}}_0(\cdot), \mathbb{B}_0(\cdot), \Pi_1(\cdot), \Pi_2(\cdot), \Pi_3(\cdot), \mathbb{Q}^2(\cdot))$ that verifies the sign condition p If there exists $\tau \in I_1 \cap I_2$ such that $X^2(\tau,i) \succeq X^1(\tau,i)$ for each $i \in

Figures (2)

  • Figure 1: The error of the Stochastic GTRDEs corresponding to System Mode 1
  • Figure 2: The error of the Stochastic GTRDEs corresponding to System Mode 2

Theorems & Definitions (13)

  • Definition 2.3
  • Theorem 2.4: *Dragan2020
  • Remark 2.5
  • Remark 2.6
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3: *Dragan2013book
  • Remark 3.4
  • Proposition 3.5
  • proof
  • ...and 3 more