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Bayesian Distributionally Robust Merton Problem with Nonlinear Wasserstein Projections

Jose Blanchet, Jiayi Cheng, Hao Liu, Yang Liu

TL;DR

The paper tackles drift-uncertainty in Merton’s continuous-time portfolio problem by introducing Distributionally Robust Bayesian Control (DRBC), which uses a single prior-level Wasserstein (and KL) ambiguity around the drift while treating volatility as known. A Sion-type minimax swap reduces DRBC to optimizing a nonlinear prior functional, with closed-form evaluation via Karatzas–Zhao, and small-radius Wasserstein perturbations are characterized through an explicit pushforward. The authors introduce nonlinear Wasserstein projections to calibrate the ambiguity radius and provide data-driven procedures to construct priors from historical data. Synthetic and real-data experiments show DRBC reduces over-pessimism relative to time-rectangular DRO and outperforms myopic DRO–Markowitz under frequent rebalancing, highlighting its practical potential in frequent-trading regimes. Overall, the work provides a tractable, learning-preserving robustness framework for continuous-time portfolio decisions under drift uncertainty with demonstrated improvements in performance and risk management.

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.

Bayesian Distributionally Robust Merton Problem with Nonlinear Wasserstein Projections

TL;DR

The paper tackles drift-uncertainty in Merton’s continuous-time portfolio problem by introducing Distributionally Robust Bayesian Control (DRBC), which uses a single prior-level Wasserstein (and KL) ambiguity around the drift while treating volatility as known. A Sion-type minimax swap reduces DRBC to optimizing a nonlinear prior functional, with closed-form evaluation via Karatzas–Zhao, and small-radius Wasserstein perturbations are characterized through an explicit pushforward. The authors introduce nonlinear Wasserstein projections to calibrate the ambiguity radius and provide data-driven procedures to construct priors from historical data. Synthetic and real-data experiments show DRBC reduces over-pessimism relative to time-rectangular DRO and outperforms myopic DRO–Markowitz under frequent rebalancing, highlighting its practical potential in frequent-trading regimes. Overall, the work provides a tractable, learning-preserving robustness framework for continuous-time portfolio decisions under drift uncertainty with demonstrated improvements in performance and risk management.

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.

Paper Structure

This paper contains 30 sections, 18 theorems, 518 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related work.
  4. Organization.
  5. Preliminaries
  6. Formulation and Main Structural Results
  7. Data-Driven Formulation and Choice of Model Parameters
  8. Synthetic Experiment
  9. Understanding Model Parameters
  10. Role of Radius
  11. Real-data Experiment
  12. Experiment Design and Data Preparation
  13. DRMV
  14. DRC
  15. DRBC
  16. ...and 15 more sections

Key Result

Lemma 1

Under Assumptions ass:utility and ass:compact_B, the map $L$ in eq:LKdef is finite for all $(k,s,y)\in(0,\infty)\times[0,T]\times\mathbb{R}^d$, continuously differentiable in $(k,s,y)$ on $(0,\infty)\times(0,T]\times\mathbb{R}^d$, and twice continuously differentiable in $(k,y)$ on $(0,\infty)\times

Figures (5)

  • Figure 1: Expectation of terminal utility gap versus $\delta$ scaling factor for DRC and DRBC in high dimensional case with Wasserstein uncertainty
  • Figure 2: Histogram of Sharpe Ratios for Bayesian Merton and DRBC
  • Figure 3: Histogram of Sharpe Ratios for DRMV with riskless asset and DRBC
  • Figure 4: Histogram of Sharpe Ratios for DRMV without riskless asset and DRBC
  • Figure 5: Histogram of Sharpe Ratios for DRC and DRBC

Theorems & Definitions (33)

  • Remark 1
  • Lemma 1
  • Theorem 1: Karatzas--Zhao KZ98's Solution
  • Theorem 2: Min-Max Equality
  • proof
  • Theorem 3: Nonlinear optimal perturbations
  • proof
  • Corollary 1: Asymptotic non-linear optimal perturbation under Wasserstein distance
  • proof
  • Remark 2: Connection to Dual Problem
  • ...and 23 more