# A Temporal Difference Method for Stochastic Continuous Dynamics

arXiv: https://arxiv.org/abs/2505.15544 · HTML: https://mlsanity.com/paper/2505.15544/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 19 of its 34 references that are in the map.
- First posted on arXiv 21 May 2025.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 19 of 19)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [Dream to Control: Learning Behaviors by Latent Imagination](https://mlsanity.com/paper/1912.01603/) (2019)
- [Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm](https://mlsanity.com/paper/1712.01815/) (2017)
- [Brax -- A Differentiable Physics Engine for Large Scale Rigid Body Simulation](https://mlsanity.com/paper/2106.13281/) (2021)
- [Policy Gradient and Actor-Critic Learning in Continuous Time and Space: Theory and Algorithms](https://mlsanity.com/paper/2111.11232/) (2021)
- [q-Learning in Continuous Time](https://mlsanity.com/paper/2207.00713/) (2022)
- [Policy Evaluation and Temporal-Difference Learning in Continuous Time and Space: A Martingale Approach](https://mlsanity.com/paper/2108.06655/) (2021)
- [Hyperparameters in Reinforcement Learning and How To Tune Them](https://mlsanity.com/paper/2306.01324/) (2023)
- [Making Deep Q-learning methods robust to time discretization](https://mlsanity.com/paper/1901.09732/) (2019)
- [Continuous-Time Model-Based Reinforcement Learning](https://mlsanity.com/paper/2102.04764/) (2021)
- [Exploratory HJB equations and their convergence](https://mlsanity.com/paper/2109.10269/) (2021)
- [Hamilton-Jacobi Deep Q-Learning for Deterministic Continuous-Time Systems with Lipschitz Continuous Controls](https://mlsanity.com/paper/2010.14087/) (2020)
- [Policy Optimization for Continuous Reinforcement Learning](https://mlsanity.com/paper/2305.18901/) (2023)
- [High-probability sample complexities for policy evaluation with linear function approximation](https://mlsanity.com/paper/2305.19001/) (2023)
- [Distributional Hamilton-Jacobi-Bellman Equations for Continuous-Time Reinforcement Learning](https://mlsanity.com/paper/2205.12184/) (2022)
- [Temporal Difference Learning with Continuous Time and State in the Stochastic Setting](https://mlsanity.com/paper/2202.07960/) (2022)

## Built on by (most cited first, 3 of 3)

- [Mean-Field PhiBE: Continuous-Time Mean-Field Reinforcement Learning from Discrete-Time Data](https://mlsanity.com/paper/2606.26498/) (2026)
- [Physics-informed Goal-Conditioned Reinforcement Learning under Hybrid Contact Dynamics](https://mlsanity.com/paper/2605.30503/) (2026)
- [Path Integral Value Matching for Linear Quadratic Stochastic Optimal Control](https://mlsanity.com/paper/2608.10777/) (2026)

## Abstract (arXiv)

For continuous systems modeled by dynamical equations such as ODEs and SDEs, Bellman's Principle of Optimality takes the form of the Hamilton-Jacobi-Bellman (HJB) equation, which provides the theoretical target of reinforcement learning (RL). Although recent advances in RL successfully leverage this formulation, the existing methods typically assume the underlying dynamics are known a priori because they need explicit access to the coefficient functions of dynamical equations to update the value function following the HJB equation. We address this inherent limitation of HJB-based RL; we propose a model-free approach still targeting the HJB equation and propose the corresponding temporal difference method. We establish exponential convergence of the idealized continuous-time dynamics and empirically demonstrate its potential advantages over transition-kernel-based formulations. The proposed formulation paves the way toward bridging stochastic control and model-free reinforcement learning.