Abstract:  The role of signatures in solving non-Markovian control problems has been increasingly recognized, particularly in areas of mathematical finance, such as optimal execution, portfolio optimization, and the valuation of American options. In this work, we study a general class of differential equations driven by stochastic rough paths, where the control impacts the system's drift. In the theoretical aspect, we demonstrate that optimal controls can be approximated using linear and deep signature functionals. This includes a stability result for controlled rough differential equations and a refined lifting result for progressively measurable processes into continuous path-functionals. Building on these theoretical insights, we have developed a practical numerical methodology based on Monte-Carlo sampling and deep learning techniques. We demonstrate the efficiency of this methodology through numerical examples, including the optimal tracking of fractional Brownian motion, for which we provide exact theoretical benchmarks.

Bio: Paul Hager recently started as a junior research group leader at TU Berlin, specializing in quantitative finance and stochastic control. He completed his PhD in 2021 and meanwhile worked as a post-doctoral researcher at HU Berlin. Paul's research focuses on modern approaches to stochastic control, particularly emplyoing signature methods. He is known for his work on "Optimal Stopping with Signatures" and the "Signature Cumulant Expansion." Currently, he is exploring the application of signature methods combined with machine learning techniques for stochastic control problems, mean-field modeling, and volatility calibration.