Sampled-Data Wasserstein Distributionally Robust Control of Multiplicative Systems: A Convex Relaxation with Performance Guarantees

Paper: arXiv 2602.04219

Authors: Chung-Han Hsieh

Abstract

This paper investigates the robust optimal control of sampled-data stochastic systems with multiplicative noise and distributional ambiguity. We consider a class of discrete-time optimal control problems where the controller \emph{jointly} selects a feedback policy and a sampling period to maximize the worst-case expected concave utility of the inter-sample growth factor. Modeling uncertainty via a Wasserstein ambiguity set, we confront the structural obstacle of~``concave-max’’ geometry arising from maximizing a concave utility against an adversarial distribution. Unlike standard convex loss minimization, the dual reformulation here requires a minimax interchange within the semi-infinite constraints, where the utility’s concavity precludes exact strong duality. To address this, we utilize a general minimax inequality to derive a tractable convex relaxation. Our approach yields a rigorous lower bound that functions as a probabilistic performance guarantee. We establish an explicit, non-asymptotic bound on the resulting duality gap, proving that the approximation error is uniformly controlled by the Lipschitz-smoothness of the stage reward and the diameter of the disturbance support. Furthermore, we introduce necessary and sufficient conditions for \emph{robust viability}, ensuring state positivity invariance across the entire ambiguity set. Finally, we bridge the gap between static optimization and dynamic performance, proving that the optimal value of the relaxation serves as a rigorous deterministic floor for the asymptotic average utility rate almost surely. The framework is illustrated on a log-optimal portfolio control problem, which serves as a canonical instance of multiplicative stochastic control.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a mathematically rigorous framework unifying Wasserstein distributionally robust control with sampled-data multiplicative systems, offering explicit duality gap bounds and asymptotic performance guarantees. However, it leans heavily theoretical, with the empirical component limited to an illustrative empirical validation on S&P 500 assets rather than an extensive backtest.

Research Flowchart

  flowchart TD
    A[Research Goal: Robust Optimal Control for Sampled-Data Systems with Multiplicative Noise] --> B[Inputs: Wasserstein Ambiguity Set, Concave Utility, Sampling Period]
    B --> C[Challenge: Dual Reformulation of Concave-Max Geometry]
    C --> D[Methodology: Minimax Inequality & Convex Relaxation]
    D --> E[Computational Process: Solve Tractable Convex Optimization]
    E --> F[Outcomes: Explicit Non-Asymptotic Duality Gap Bound & Viability Conditions]
    F --> G[Validation: Guaranteed Asymptotically Floor Rate in Portfolio Control]

Cited References (170)

1. The Price of Robustness (2004)
2. Quantifying Distributional Model Risk via Optimal Transport (2019)
3. A Framework for Stabilization of Nonlinear Sampled-Data Systems based on Their Approximate Discrete-Time Models (2004)
4. Discrete-Time Controlled Markov Processes with Average Cost Criterion: A Survey (1993)
5. Control of Markov Chains with Long-Run Average Cost Criterion (1988)
6. Risk-Sensitive Optimal Control (1990)
7. Risk-Sensitive Control on an Infinite Time Horizon (1995)
8. Wasserstein Distributionally Robust Control of Partially Observable Linear Stochastic Systems (2024)

4 of 170 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.