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 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.