# Dual Approaches to Stochastic Control via SPDEs and the Pathwise Hopf Formula

arXiv: https://arxiv.org/abs/2604.08155 · HTML: https://mlsanity.com/paper/2604.08155/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 14 of its 36 references that are in the map.
- First posted on arXiv 9 Apr 2026.

## 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, 14 of 14)

- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [A Tour of Reinforcement Learning: The View from Continuous Control](https://mlsanity.com/paper/1806.09460/) (2018)
- [Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space](https://mlsanity.com/paper/2005.05409/) (2020)
- [Deep Learning Approximation for Stochastic Control Problems](https://mlsanity.com/paper/1611.07422/) (2016)
- [Recent Developments in Machine Learning Methods for Stochastic Control and Games](https://mlsanity.com/paper/2303.10257/) (2023)
- [A Neural Network Approach for Stochastic Optimal Control](https://mlsanity.com/paper/2209.13104/) (2022)
- [Perspectives on characteristics based curse-of-dimensionality-free numerical approaches for solving Hamilton-Jacobi equations](https://mlsanity.com/paper/1711.03314/) (2017)
- [Solving Time-Continuous Stochastic Optimal Control Problems: Algorithm Design and Convergence Analysis of Actor-Critic Flow](https://mlsanity.com/paper/2402.17208/) (2024)
- [Algorithm for Overcoming the Curse of Dimensionality for State-dependent Hamilton-Jacobi equations](https://mlsanity.com/paper/1704.02524/) (2017)
- [A deep primal-dual BSDE method for optimal stopping problems](https://mlsanity.com/paper/2409.06937/) (2024)
- [Stochastic control with rough paths](https://mlsanity.com/paper/1303.7160/) (2013)
- [UVIP: Model-Free Approach to Evaluate Reinforcement Learning Algorithms](https://mlsanity.com/paper/2105.02135/) (2021)
- [Adversarial Reinforcement Learning: A Duality-Based Approach To Solving Optimal Control Problems](https://mlsanity.com/paper/2506.00801/) (2025)
- [DeepMartingale: Duality of the Optimal Stopping Problem with Expressivity and High-Dimensional Hedging](https://mlsanity.com/paper/2510.13868/) (2025)

## Abstract (arXiv)

We develop dual approaches for continuous-time stochastic control problems, enabling the computation of robust dual bounds in high-dimensional state and control spaces. Building on the dual formulation proposed in [L. C. G. Rogers, SIAM Journal on Control and Optimization, 46 (2007), pp. 1116--1132], we first formulate the inner optimization problem as a stochastic partial differential equation (SPDE); the expectation of its solution yields the dual bound. Curse-of-dimensionality-free methods are proposed based on the Pontryagin maximum principle and the generalized Hopf formula. In the process, we prove the generalized Hopf formula, first introduced as a conjecture in [Y. T. Chow, J. Darbon, S. Osher, and W. Yin, Journal of Computational Physics 387 (2019), pp. 376--409], under mild conditions. Numerical experiments demonstrate that our dual approaches effectively complement primal methods, including the deep BSDE method for solving high-dimensional PDEs and the deep actor-critic method in reinforcement learning.