Non-concave stochastic optimal control in finite discrete time under model uncertainty

Paper: arXiv 2404.05230

Authors: Ariel Neufeld, Julian Sester

Abstract

In this article we present a general framework for non-concave robust stochastic control problems under model uncertainty in a discrete time finite horizon setting. Our framework allows to consider a variety of different path-dependent ambiguity sets of probability measures comprising, as a natural example, the ambiguity set defined via Wasserstein-balls around path-dependent reference measures with path-dependent radii, as well as parametric classes of probability distributions. We establish a dynamic programming principle which allows to derive both optimal control and worst-case measure by solving recursively a sequence of one-step optimization problems. Moreover, we derive upper bounds for the difference of the values of the robust and non-robust stochastic control problem in the Wasserstein uncertainty and parameter uncertainty case. As a concrete application, we study the robust hedging problem of financial derivatives under an asymmetric (and non-convex) loss function accounting for different preferences of sell- and buy side when it comes to the hedging of financial derivatives. As our entirely data-driven ambiguity set of probability measures, we consider Wasserstein-balls around the empirical measure derived from real financial data. We demonstrate that during adverse scenarios such as a financial crisis, our robust approach outperforms typical model-based hedging strategies such as the classical Delta-hedging strategy as well as the hedging strategy obtained in the non-robust setting with respect to the empirical measure and therefore overcomes the problem of model misspecification in such critical periods.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework for non-concave stochastic optimal control under path-dependent model uncertainty using dynamic programming. It balances theoretical heavy lifting with a practical, data-driven application to asymmetric derivative hedging using empirical Wasserstein balls, outperforming traditional delta-hedging during market stress.

Research Flowchart

  flowchart TD
    A[Goal: Robust Non-Concave Control<br/>under Model Uncertainty] --> B[Define Path-Dependent Ambiguity Sets<br/>e.g., Wasserstein-balls & Parametric]
    B --> C[Derive Recursive Dynamic Programming Principle<br/>for One-Step Optimization]
    
    D[Real Financial Market Data] --> E[Application: Empirical Derivatives Hedging<br/>with Asymmetric Loss]
    C --> E
    
    E --> F[Establish Quantitative Bounds for<br/>Robust vs. Non-Robust Value Gap]
    E --> G[Outperforms Classical Delta-Hedging &<br/>Non-Robust Strategies in Adverse Scenarios]

Cited References (215)

1. Advances in prospect theory: Cumulative representation of uncertainty (1992)
2. Optimal delta hedging for options (2017)
3. Kernel Methods for Time Series Classification and Regression. (2018)
4. Central limit theorems for the Wasserstein distance between the empirical and the true distributions (1999)
5. Adam: A method for stochastic optimization (2014) — link
6. Nonconcave Robust Utility Maximization under Projective Determinacy — → 2403.11824
7. Distributionally robust control of constrained stochastic systems (2015)
8. Sinkhorn distributionally robust optimization (2021) — link

6 of 215 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.