Continuous-time optimal investment with portfolio constraints: a reinforcement learning approach

TL;DR

This paper develops a reinforcement learning framework for continuous-time portfolio optimization with constraints, deriving explicit policies and demonstrating how exploration influences wealth distribution and investment opportunities.

Contribution

It introduces a novel RL approach for constrained continuous-time investment, providing explicit Gaussian and truncated Gaussian policies, and establishes a policy improvement theorem.

Findings

Optimal policies are Gaussian or truncated Gaussian.

Exploration increases wealth dispersion and tail heaviness.

Constraints reduce exploration costs significantly.

Abstract

In a reinforcement learning (RL) framework, we study the exploratory version of the continuous time expected utility (EU) maximization problem with a portfolio constraint that includes widely-used financial regulations such as short-selling constraints and borrowing prohibition. The optimal feedback policy of the exploratory unconstrained classical EU problem is shown to be Gaussian. In the case where the portfolio weight is constrained to a given interval, the corresponding constrained optimal exploratory policy follows a truncated Gaussian distribution. We verify that the closed form optimal solution obtained for logarithmic utility and quadratic utility for both unconstrained and constrained situations converge to the non-exploratory expected utility counterpart when the exploration weight goes to zero. Finally, we establish a policy improvement theorem and devise an implementable reinforcement learning algorithm by casting the optimal problem in a martingale framework. Our numerical examples show that exploration leads to an optimal wealth process that is more dispersedly distributed with heavier tail compared to that of the case without exploration. This effect becomes less significant as the exploration parameter is smaller. Moreover, the numerical implementation also confirms the intuitive understanding that a broader domain of investment opportunities necessitates a higher exploration cost. Notably, when subjected to both short-selling and money borrowing constraints, the exploration cost becomes negligible compared to the unconstrained case.