Portfolio choice, portfolio liquidation, and portfolio transition under drift uncertainty

TL;DR

This paper develops models for optimal portfolio choice, liquidation, and transition under return uncertainty by integrating Bayesian learning with dynamic programming, extending classical results to more complex, realistic scenarios.

Contribution

It introduces a novel framework combining Bayesian learning and PDEs to address portfolio decisions under return uncertainty, including liquidity considerations.

Findings

Recover classical Merton results within the new framework

Extend to Almgren-Chriss type models with liquidity and uncertainty

Provide PDE-based solutions for complex portfolio problems

Abstract

This paper presents several models addressing optimal portfolio choice, optimal portfolio liquidation, and optimal portfolio transition issues, in which the expected returns of risky assets are unknown. Our approach is based on a coupling between Bayesian learning and dynamic programming techniques that leads to partial differential equations. It enables to recover the well-known results of Karatzas and Zhao in a framework \`a la Merton, but also to deal with cases where martingale methods are no longer available. In particular, we address optimal portfolio choice, portfolio liquidation, and portfolio transition problems in a framework \`a la Almgren-Chriss, and we build therefore a model in which the agent takes into account in his decision process both the liquidity of assets and the uncertainty with respect to their expected return.