Linear Quadratic Control with Non-Markovian and Non-Semimartingale Noise Models

TL;DR

This paper extends linear quadratic control theory to non-Markovian, non-semimartingale noise models using rough path theory, enabling optimal control and estimation in more complex stochastic environments.

Contribution

It introduces a novel framework employing rough path theory for LQG problems with irregular, non-Markovian noise, broadening the applicability of control methods.

Findings

Derived optimal control strategies using rough path signatures

Formulated state estimation under irregular noise models

Extended LQG framework beyond classical stochastic calculus

Abstract

The standard linear quadratic Gaussian (LQG) framework assumes a Brownian noise process and relies on classical stochastic calculus tools, such as those based on It\^o calculus. In this paper, we solve a generalized linear quadratic optimal control problem where the process and measurement noises can be non-Markovian and non-semimartingale stochastic processes with sample paths that have low H\"older regularity. Since these noise models do not, in general, permit the use of the standard It\^o calculus, we employ rough path theory to formulate and solve the problem. By leveraging signature representations and controlled rough paths, we derive the optimal state estimation and control strategies.