Paper: arXiv 2512.14991

Authors: Hanqing Jin, Renyuan Xu, Yanzhao Yang

Abstract

We study reinforcement learning for controlled diffusion processes with unbounded continuous state spaces, bounded continuous actions, and polynomially growing rewards: settings that arise naturally in finance, economics, and operations research. To overcome the challenges of continuous and high-dimensional domains, we introduce a model-based algorithm that adaptively partitions the joint state-action space. The algorithm maintains estimators of drift, volatility, and rewards within each partition, refining the discretization whenever estimation bias exceeds statistical confidence. This adaptive scheme balances exploration and approximation, enabling efficient learning in unbounded domains. Our analysis establishes regret bounds that depend on the problem horizon, state dimension, reward growth order, and a newly defined notion of zooming dimension tailored to unbounded diffusion processes. The bounds recover existing results for bounded settings as a special case, while extending theoretical guarantees to a broader class of diffusion-type problems. Finally, we validate the effectiveness of our approach through numerical experiments, including applications to high-dimensional problems such as multi-asset mean-variance portfolio selection.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced stochastic calculus and high-dimensional adaptive partitioning theory with complex regret bound derivations, yet it includes numerical experiments (e.g., multi-asset portfolio selection) that demonstrate practical feasibility.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>RL for Stochastic Control<br>in Unbounded Diffusion Processes"] --> B["Methodology:<br>Model-Based Adaptive Partitioning"]
  B --> C["Input Data:<br>Unbounded Continuous States<br>Bounded Actions & Polynomial Rewards"]
  C --> D["Computational Process:<br>Estimate Drift, Volatility & Rewards<br>Refine Partitions on Bias Threshold"]
  D --> E["Outcome:<br>Regret Bounds with<br>Zooming Dimension &<br>Validation via Portfolio Selection"]