Bayesian Distributionally Robust Merton Problem with Nonlinear Wasserstein Projections

2025-12-01

AI Summary

Model uncertainty arising from inaccurate drift estimation in continuous-time portfolio selection undermines robustness and performance. Method: We propose a Bayesian distributionally robust optimization (DRO) framework that constructs a single Wasserstein ambiguity set over the drift prior—replacing conventional time-rectangular robust control to mitigate excessive conservatism. A nonlinear Wasserstein projection calibrates the ambiguity radius, balancing learning adaptability with robustness while ensuring asymptotically optimal characterization of the worst-case prior. The approach integrates the Bayesian Merton model, minimax interchange techniques, and a Karatzas–Zhao-type closed-form solution. Results: Empirical and synthetic experiments demonstrate that our method significantly alleviates pessimistic bias compared to classical DRO and myopic DRO-Markowitz strategies, yielding superior risk-adjusted returns—particularly under high-frequency rebalancing.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We revisit Merton's continuous-time portfolio selection through a data-driven, distributionally robust lens. Our aim is to tap the benefits of frequent trading over short horizons while acknowledging that drift is hard to pin down, whereas volatility can be screened using realized or implied measures for appropriately selected assets. Rather than time-rectangular distributional robust control -- which replenishes adversarial power at every instant and induces over-pessimism -- we place a single ambiguity set on the drift prior within a Bayesian Merton model. This prior-level ambiguity preserves learning and tractability: a minimax swap reduces the robust control to optimizing a nonlinear functional of the prior, enabling Karatzas and Zhao cite{KZ98}-type's closed-form evaluation for each candidate prior. We then characterize small-radius worst-case priors under Wasserstein uncertainty via an explicit asymptotically optimal pushforward of the nominal prior, and we calibrate the ambiguity radius through a nonlinear Wasserstein projection tailored to the Merton functional. Synthetic and real-data studies demonstrate reduced pessimism relative to DRC and improved performance over myopic DRO-Markowitz under frequent rebalancing.

Problem

• Addresses portfolio selection under drift uncertainty in Merton's model
• Proposes Bayesian robust control with a single prior ambiguity set
• Calibrates ambiguity radius via nonlinear Wasserstein projection for performance

Innovation

• Bayesian Merton model with drift prior ambiguity
• Nonlinear Wasserstein projection for calibration
• Closed-form evaluation via minimax swap reduction