A dynamic programming principle for multiperiod control problems with bicausal constraints

Paper: arXiv 2410.23927

Authors: Ruslan Mirmominov, Johannes Wiesel

Abstract

We consider multiperiod stochastic control problems with non-parametric uncertainty on the underlying probabilistic model. We derive a new metric on the space of probability measures, called the adapted $(p, \infty)$–Wasserstein distance $\mathcal{AW}_p^\infty$ with the following properties: (1) the adapted $(p, \infty)$–Wasserstein distance generates a topology that guarantees continuity of stochastic control problems and (2) the corresponding $\mathcal{AW}_p^\infty$-distributionally robust optimization (DRO) problem can be computed via a dynamic programming principle involving one-step Wasserstein-DRO problems. If the cost function is semi-separable, then we further show that a minimax theorem holds, even though balls with respect to $\mathcal{AW}_p^\infty$ are neither convex nor compact in general. We also derive first-order sensitivity results.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical contribution by introducing a novel adapted Wasserstein distance metric that allows for a dynamic programming principle in multiperiod DRO problems. However, it is purely theoretical and lacks empirical validation, backtesting, or computational implementations.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve multiperiod stochastic control under non-parametric model uncertainty] --> B[Data/Inputs: Non-parametric probability measures & Semi-separable cost functions]
    B --> C[Methodology: Define Adapted Wasserstein Distance AW_p^∞ to capture information flow]
    C --> D[Computational Process: Formulate AW_p^∞ Distributionally Robust Optimization DRO]
    D --> E[Computational Process: Reduce DRO to nested one-step Wasserstein DRO problems]
    E --> F[Outcomes: Dynamic Programming Principle & Topology guaranteeing problem continuity]
    E --> G[Outcomes: Minimax Theorem & First-Order Sensitivity Analysis]