Advancing Optimal Stochastic Control with Signatures

Hager Paul P., Humboldt University of Berlin
Bayer Christian, WIAS Berlin
Riedel Sebastian, University of Hagen
Nauen Tobias, German Research Center for Artificial Intelligence

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 refined lifting result for progressively measurable processes into continuous path-functionals, in addition to implementing a robust stability result for rough differential equations from Diehl et al. 2017. 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. My talk will begin with an introduction to the essential basics of rough paths and their signatures.

Area: CS42 - Rough paths and data science (Christian Bayer, Paul Hager and Sebastian Riedel)

Keywords: stochastic control, rough path, signature, deep learning, rough differential equations, optimal tracking, fractional Brownian motion

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