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Robustness of optimal control for controlled regime-switching diffusions with incorrect models

Somnath Pradhan, Dinesh Rathia

Abstract

This paper investigates the robustness of stochastic optimal control for controlled regime switching diffusions. We consider systems driven by both continuous fluctuations and discrete regime changes, allowing for model misspecification in both the diffusion and switching components. Within a unified framework, we study four classical cost formulations finite horizon, infinite-horizon discounted and ergodic costs, and the exit time cost, and establish continuity of value functions and robustness of optimal controls. Specifically, we show that as a sequence of approximating regime switching models converges to the true model, the associated value functions and optimal policies converge as well, ensuring vanishing performance loss. The analysis relies on the regularity of the solution to the associated weakly coupled HJB systems, and their stochastic representation. The results extend the robustness framework developed for diffusion processes to a significantly broader class of hybrid systems with interacting continuous and discrete dynamics.

Robustness of optimal control for controlled regime-switching diffusions with incorrect models

Abstract

This paper investigates the robustness of stochastic optimal control for controlled regime switching diffusions. We consider systems driven by both continuous fluctuations and discrete regime changes, allowing for model misspecification in both the diffusion and switching components. Within a unified framework, we study four classical cost formulations finite horizon, infinite-horizon discounted and ergodic costs, and the exit time cost, and establish continuity of value functions and robustness of optimal controls. Specifically, we show that as a sequence of approximating regime switching models converges to the true model, the associated value functions and optimal policies converge as well, ensuring vanishing performance loss. The analysis relies on the regularity of the solution to the associated weakly coupled HJB systems, and their stochastic representation. The results extend the robustness framework developed for diffusion processes to a significantly broader class of hybrid systems with interacting continuous and discrete dynamics.

Paper Structure

This paper contains 14 sections, 29 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Description of the problem
  3. Assumptions:
  4. Cost criteria:
  5. Approximating Control Regime-Switching Diffusion Process:
  6. Continuity and Robustness problems
  7. Analysis of Discounted Cost Criterion
  8. Analysis of Ergodic Cost
  9. Analysis under Lyapunov stability
  10. Finite Horizon Cost
  11. Control up to an Exit Time
  12. Example
  13. Conclusion
  14. Verification Theorem

Key Result

Theorem 3.1

Suppose Assumptions A1–A5 hold. Then the optimal discounted cost function $V_{\alpha}$, defined in OPDcost, is the unique solution in $\mathcal{C}^2({\mathds{R}^{d}} \times \mathbb{S}) \cap \mathcal{C}_b({\mathds{R}^{d}} \times \mathbb{S})$ that satisfies the Hamilton–Jacobi–Bellman (HJB) equation Moreover, $v^*\in \mathfrak U_{\mathsf{sm}}$ is $\alpha$-discounted optimal control if and only if i

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark 3.1
  • Theorem 4.1
  • ...and 45 more