# Mean--Variance Portfolio Selection by Continuous-Time Reinforcement Learning: Algorithms, Regret Analysis, and Empirical Study

arXiv: https://arxiv.org/abs/2412.16175 · HTML: https://mlsanity.com/paper/2412.16175/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Adversarial robustness and RL in finance, one of 15 directions in Deep reinforcement learning methods.
- First posted on arXiv 8 Dec 2024.

## Placed in

- Deep reinforcement learning methods / [Adversarial robustness and RL in finance](https://mlsanity.com/deeprl/adversarial-robustness-and-rl-in-finance/)

## Built on by (most cited first, 8 of 8)

- [ART for Diffusion Sampling: A Reinforcement Learning Approach to Timestep Schedule](https://mlsanity.com/paper/2601.18681/) (2026)
- [Data-Driven Exploration for a Class of Continuous-Time Indefinite Linear--Quadratic Reinforcement Learning Problems](https://mlsanity.com/paper/2507.00358/) (2025)
- [Amortized Guidance for Image Inpainting with Pretrained Diffusion Models](https://mlsanity.com/paper/2605.13010/) (2026)
- [Diffusion Models for Adaptive Sequential Data Generation](https://mlsanity.com/paper/2606.06007/) (2026)
- [Factor-Based Conditional Diffusion Model for Contextual Portfolio Optimization](https://mlsanity.com/paper/2509.22088/) (2025)
- [ART for Diffusion Sampling: Continuous-Time Control and Actor-Critic Learning](https://mlsanity.com/paper/2607.02137/) (2026)
- [Dynamic mean-variance portfolio selection with no-shorting constraints and unknown investment opportunity sets](https://mlsanity.com/paper/2607.16625/) (2026)
- [Learning to Solve Stochastic Controls with Unknown Drifts and Running Rewards: Theory, Algorithms and Convergence](https://mlsanity.com/paper/2609.14972/) (2026)

## Abstract (arXiv)

We study continuous-time mean--variance portfolio selection in markets where stock prices are diffusion processes driven by observable factors that are also diffusion processes, yet the coefficients of these processes are unknown. Based on the recently developed reinforcement learning (RL) theory for diffusion processes, we present a general data-driven RL approach that learns the pre-committed investment strategy directly without attempting to learn or estimate the market coefficients. For multi-stock Black--Scholes markets without factors, we further devise an algorithm and prove its performance guarantee by deriving a sublinear regret bound in terms of the Sharpe ratio. We then carry out an extensive empirical study implementing this algorithm to compare its performance and trading characteristics, evaluated under a host of common metrics, with a large number of widely employed portfolio allocation strategies on S&P 500 constituents. The results demonstrate that the proposed continuous-time RL strategy is consistently among the best, especially in a volatile bear market, and decisively outperforms the model-based continuous-time counterparts by significant margins.