# The Pontryagin maximum principle and Q-functions in rough environments

arXiv: https://arxiv.org/abs/2601.05354 · HTML: https://mlsanity.com/paper/2601.05354/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jan 2026.

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

- [Unified Optimality Conditions for Stochastic Optimal Control in the Rough Path and Ito Frameworks](https://mlsanity.com/paper/2609.38395/) (2026)

## Abstract (arXiv)

We derive the Pontryagin maximum principle and Q-functions for the relaxed control of noisy rough differential equations. Our main tool is the development of a novel differentiation procedure along `spike variation' perturbations of the optimal state-control pair. We then exploit our development of the infinitesimal Q-function (also known as the q-function) to derive a policy improvement algorithm for settings with entropic cost constraints.