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

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.

💡 Research Summary

The paper tackles the fundamental limitation of the classical Linear‑Quadratic‑Gaussian (LQG) framework, which relies on Brownian motion and Itô calculus, by extending optimal control to environments where the process and measurement noises are both non‑Markovian and non‑semimartingale with low Hölder regularity (H∈(1/3,1]). Such noises include fractional Brownian motion with H<½, stable Lévy processes, and shot‑noise models in quantum systems. Because Itô integrals are undefined for these signals, the authors adopt rough path theory, lifting each noise trajectory to a geometric p‑rough path (p∈