Duality and Policy Evaluation in Distributionally Robust Bayesian Diffusion Control

Paper: arXiv 2506.19294

Authors: Jose Blanchet, Jiayi Cheng, Hao Liu, Yang Liu

Abstract

We consider a Bayesian diffusion control problem of expected terminal utility maximization. The controller imposes a prior distribution on the unknown drift of an underlying diffusion. The Bayesian optimal control, tracking the posterior distribution of the unknown drift, can be characterized explicitly. However, in practice, the prior will generally be incorrectly specified, and the degree of model misspecification can have a significant impact on policy performance. To mitigate this and reduce overpessimism, we introduce a distributionally robust Bayesian control (DRBC) formulation in which the controller plays a game against an adversary who selects a prior in divergence neighborhood of a baseline prior. The adversarial approach has been studied in economics and efficient algorithms have been proposed in static optimization settings. We develop a strong duality result for our DRBC formulation. Combining these results together with tools from stochastic analysis, we are able to derive a loss that can be efficiently trained (as we demonstrate in our numerical experiments) using a suitable neural network architecture. As a result, we obtain an effective algorithm for computing the DRBC optimal strategy. The methodology for computing the DRBC optimal strategy is greatly simplified, as we show, in the important case in which the adversary chooses a prior from a Kullback-Leibler distributional uncertainty set.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 5.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents advanced stochastic analysis, duality theory, and PDE-like formulations for a continuous-time control problem, indicating high mathematical complexity. While it includes theoretical results (e.g., sample complexity rates) and a neural network architecture for learning, the empirical validation is based on simulations rather than live trading data or extensive backtests, placing it in the middle of empirical rigor.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Distributionally Robust<br>Bayesian Diffusion Control"] --> B["Key Methodology:<br>Duality & Stochastic Analysis<br>for DRBC Formulation"]
  B --> C["Data/Input:<br>Baseline Prior &<br>Adversarial Uncertainty Set<br>e.g., KL Divergence"]
  C --> D["Computational Process:<br>Derive Explicit Loss &<br>Train Neural Network"]
  D --> E["Key Findings:<br>Effective Algorithm for<br>Optimal DRBC Strategy"]
  E --> F["Outcome:<br>Reduced Overpessimism &<br>Improved Robustness"]

Cited References (147)

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2. A well-conditioned estimator for large-dimensional covariance matrices (2004)
3. On Nesting Monte Carlo Estimators (2018)
4. Rates of Convergence and CLTs for Subcanonical Debiased MLMC (2018)
5. Robust Multi-Objective Bayesian Optimization under Input Noise (2022)
6. Recent Advances in Reinforcement Learning in Finance (2023)
7. Brownian Motion and Stochastic Calculus (1991)
8. Lectures on Stochastic Programming: Modeling and Theory (2009)

4 of 147 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.