# Unified Optimality Conditions for Stochastic Optimal Control in the Rough Path and Ito Frameworks

arXiv: https://arxiv.org/abs/2609.38395 · HTML: https://mlsanity.com/paper/2609.38395/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 30 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Adjoint Matching through the Lens of the Stochastic Maximum Principle in Optimal Control](https://mlsanity.com/paper/2604.08580/) (2026)
- [Sequential Convex Programming For Non-Linear Stochastic Optimal Control](https://mlsanity.com/paper/2009.05182/) (2020)
- [Stochastic control with rough paths](https://mlsanity.com/paper/1303.7160/) (2013)
- [Rough Stochastic Pontryagin Maximum Principle and an Indirect Shooting Method](https://mlsanity.com/paper/2502.06726/) (2025)
- [Pontryagin Maximum Principle for rough stochastic systems and pathwise stochastic control](https://mlsanity.com/paper/2503.22959/) (2025)
- [The Pontryagin maximum principle and Q-functions in rough environments](https://mlsanity.com/paper/2601.05354/) (2026)
- [Duality for Stochastic Control with non-Markovian Random Coefficients](https://mlsanity.com/paper/2609.05101/) (2026)

## Abstract (arXiv)

Stochastic differential equations (SDEs) can be studied via Ito calculus and rough path theory. For stochastic optimal control, these two frameworks give distinct Pontryagin Maximum Principle (PMP) optimality conditions with forward-backward SDEs (FBSDEs) or rough differential equations. We show that the adjoint equations of the Ito and rough PMPs are connected via the conditional expectation pt^textIto=mathbbE[ptrough mid Ft], where Ft represents information available at time t. First, we derive a rough stochastic PMP for problems with adapted controls that does not use FBSDEs. Its proof extends the rough stochastic PMP over deterministic controls by considering stochastic needle variations. Second, we derive a unified PMP connecting the Ito and rough PMPs, using Ito-Stratonovich conversion formulas and duality identities between the forward tangent and backward adjoint SDEs. As a first application, we rederive the adjoint matching method for fine-tuning generative models. As a second application, we propose an indirect shooting method for a class of feedback problems. Overall, these results give a new conditional bridge connecting two popular frameworks for stochastic optimal control.