Solving Linear-Quadratic Stochastic Control Problems with Signatures

TL;DR

This paper introduces a signature-driven numerical method for solving multi-dimensional linear-quadratic stochastic control problems, transforming them into convex quadratic optimizations and demonstrating effective approximations with low truncation levels.

Contribution

The paper develops a novel signature-based numerical scheme that converts complex stochastic control problems into deterministic convex quadratic problems, with proven convergence and practical efficiency.

Findings

The signature approach accurately approximates the value function with low truncation levels.

The method transforms stochastic control problems into deterministic quadratic optimizations.

Numerical experiments show high accuracy with small signature truncations.

Abstract

We study a signature-driven numerical scheme to solve multi-dimensional linear-quadratic (LQ) stochastic control problems. Using that linear signature functionals are dense in the natural class of admissible controls, we show that our approach turns the original LQ problem into a deterministic convex quadratic polynomial optimisation. To underpin a numerical approach based on truncated signatures, we prove that the problem's value function can be approximated by finite-dimensional polynomial approximations when the truncation levels are chosen sufficiently high. Remarkably, our numerical experiments show very decent accuracy already for small truncation levels. Key tools for our analysis are (i) the algebraic representation of controlled stochastic differential equations and the associated cost function as linear functionals of the path signatures of the driving noise, (ii) the convergence of the truncated linear functionals, and (iii) the density of signature controls.