Robustness of Optimal Controlled Diffusions with Near-Brownian Noise via Rough Paths Theory

TL;DR

This paper demonstrates that optimal control solutions for idealized Brownian-driven systems remain nearly optimal when the noise is approximated by more realistic, near-Brownian processes, using rough paths theory for robustness analysis.

Contribution

It introduces a robustness theorem for controlled diffusions driven by near-Brownian noise, extending control theory to more realistic stochastic models via rough paths.

Findings

Optimal controls for Brownian models are nearly optimal under near-Brownian noise.

Rough paths theory provides a pathwise framework for robustness analysis.

The results apply within Lipschitz continuous control policies.

Abstract

In this article we show a robustness theorem for controlled stochastic differential equations driven by approximations of Brownian motion. Often, Brownian motion is used as an idealized model of a diffusion where approximations such as Wong-Zakai, Karhnen-Lo\`eve or fractional Brownian motion are often seen as more physical. However, there has been extensive literature on solving control problems driven by Brownian motion and little on control problems driven by more realistic models that are only approximately Brownian. The question of robustness naturally arises from such approximations. We show robustness using rough paths theory, which allows for a pathwise theory of stochastic differential equations. To this end, in particular, we show that within the class of Lipschitz continuous control policies, an optimal solution for the Brownian idealized model is near optimal for a true system driven by a non-Brownian (but near-Brownian) noise.