Path-Dependent SDEs: Solutions and Parameter Estimation

TL;DR

This paper introduces a novel method called ESMM for accurately estimating parameters of path-dependent SDEs using path signatures, providing a general framework for modeling and inference in complex stochastic systems.

Contribution

It develops the Expected Signature Matching Method (ESMM) for consistent parameter estimation in signature SDEs, advancing beyond traditional parametric approaches.

Findings

ESMM accurately infers drift and diffusion parameters from observed trajectories.

Theoretical proof of consistency for the ESMM approach.

Empirical simulations demonstrate the effectiveness of ESMM in various scenarios.

Abstract

We develop a consistent method for estimating the parameters of a rich class of path-dependent SDEs, called signature SDEs, which can model general path-dependent phenomena. Path signatures are iterated integrals of a given path with the property that any sufficiently nice function of the path can be approximated by a linear functional of its signatures. This is why we model the drift and diffusion of our signature SDE as linear functions of path signatures. We provide conditions that ensure the existence and uniqueness of solutions to a general signature SDE. We then introduce the Expected Signature Matching Method (ESMM) for linear signature SDEs, which enables inference of the signature-dependent drift and diffusion coefficients from observed trajectories. Furthermore, we prove that ESMM is consistent: given sufficiently many samples and Picard iterations used by the method, the parameters estimated by the ESMM approach the true parameter with arbitrary precision. Finally, we demonstrate on a variety of empirical simulations that our ESMM accurately infers the drift and diffusion parameters from observed trajectories. While parameter estimation is often restricted by the need for a suitable parametric model, this work makes progress toward a completely general framework for SDE parameter estimation, using signature terms to model arbitrary path-independent and path-dependent processes.