# Pathwise Relaxed Optimal Control of Rough Differential Equations

arXiv: https://arxiv.org/abs/2402.17900 · HTML: https://mlsanity.com/paper/2402.17900/

## Key facts

- Cited by 1 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 7 of its 41 references that are in the map.
- First posted on arXiv 27 Feb 2024.

## 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, 7 of 7)

- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [A unified view of entropy-regularized Markov decision processes](https://mlsanity.com/paper/1705.07798/) (2017)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [A Neural RDE approach for continuous-time non-Markovian stochastic control problems](https://mlsanity.com/paper/2306.14258/) (2023)
- [Solving non-Markovian Stochastic Control Problems driven by Wiener Functionals](https://mlsanity.com/paper/2003.06981/) (2020)
- [Pathwise stochastic control and a class of stochastic partial differential equations](https://mlsanity.com/paper/2301.09214/) (2023)

## Built on by (most cited first, 1 of 1)

- [Adaptive Learning via Off-Model Training and Importance Sampling for Fully Non-Markovian Optimal Stochastic Control. Complete version](https://mlsanity.com/paper/2604.13147/) (2026)

## Abstract (arXiv)

This note lays part of the theoretical ground for a definition of differential systems modeling reinforcement learning in continuous time non-Markovian rough environments. Specifically we focus on optimal relaxed control of rough equations (the term relaxed referring to the fact that controls have to be considered as measure valued objects). With reinforcement learning in view, our reward functions encompass forms that involve an entropy-type term favoring exploration. In this context, our contribution focuses on a careful definition of the corresponding relaxed Hamilton-Jacobi-Bellman (HJB)-type equation. A substantial part of our endeavor consists in a precise definition of the notion of test function and viscosity solution for the rough relaxed PDE obtained in this framework. Note that this task is often merely sketched in the rough viscosity literature, in spite of the fact that it gives a proper meaning to the differential system at stake. In the last part of the paper we prove that the natural value function in our context solves a relaxed rough HJB equation in the viscosity sense.