Signature moments to characterize laws of stochastic processes

TL;DR

This paper introduces robust signature moments to characterize the laws of stochastic processes, enabling a new metric and efficient kernel-based computations, with applications in non-parametric two-sample tests.

Contribution

It develops a novel approach using signature moments for path-valued random variables, extending moment-based characterization to stochastic processes.

Findings

Defined a maximum mean discrepancy type metric for stochastic process laws

Introduced a kernelized version using signature kernels for efficient computation

Demonstrated a non-parametric two-sample test for stochastic processes

Abstract

The sequence of moments of a vector-valued random variable can characterize its law. We study the analogous problem for path-valued random variables, that is stochastic processes, by using so-called robust signature moments. This allows us to derive a metric of maximum mean discrepancy type for laws of stochastic processes and study the topology it induces on the space of laws of stochastic processes. This metric can be kernelized using the signature kernel which allows to efficiently compute it. As an application, we provide a non-parametric two-sample hypothesis test for laws of stochastic processes.