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

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 Itô 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 Hölder regularity. Since these noise models do not, in general, permit the use of the standard Itô 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.