# Optimal control of SDEs with merely measurable drift: an HJB approach

arXiv: https://arxiv.org/abs/2509.01054 · HTML: https://mlsanity.com/paper/2509.01054/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Sep 2025.

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

- [Mollified Value Learning](https://mlsanity.com/paper/2602.23280/) (2026)

## Abstract (arXiv)

We investigate an optimal control problem for a diffusion whose drift and running cost are merely measurable in the state variable. Such low regularity rules out the use of Pontryagin's maximum principle and also invalidates the standard proof of the Bellman principle of optimality. We address these difficulties by analyzing the associated Hamilton-Jacobi-Bellman (HJB) equation. Working in a weak formulation of admissible controls, we first establish the state-equation solvability and Krylov estimates needed to make the control problem well defined. Using PDE techniques together with a policy iteration scheme, we prove that the HJB equation admits a unique strong solution, and this solution coincides with the value function of the control problem. Based on this identification, we establish a verification theorem and recover the Bellman optimality principle without imposing any additional smoothness assumptions. We further investigate a mollification scheme depending on a parameter varepsilon > 0. It turns out that the smoothed value functions Vvarepsilon may fail to converge to the original value function V as varepsilon → 0, and we provide an explicit counterexample. To resolve this, we identify a structural condition on the control set. When the control set is countable, convergence Vvarepsilon → V holds locally uniformly.