Path-dependent processes from signatures

TL;DR

This paper introduces explicit series expansions for stochastic path-dependent integral equations using path signatures, enabling better understanding and approximation of complex stochastic processes like fractional Brownian motion.

Contribution

It provides a novel framework that expresses solutions of stochastic path-dependent equations in terms of path signatures, applicable to a broad class including fractional Brownian motion.

Findings

Series expansions facilitate disentangling infinite-dimensional structures.

Framework applies to stochastic Volterra and delay equations.

Numerical illustrations demonstrate practical approximation schemes.

Abstract

We provide explicit series expansions to certain stochastic path-dependent integral equations in terms of the path signature of the time augmented driving Brownian motion. Our framework encompasses a large class of stochastic linear Volterra and delay equations and in particular the fractional Brownian motion with a Hurst index $H \in (0, 1)$. Our expressions allow to disentangle an infinite dimensional Markovian structure and open the door to straightforward and simple approximation schemes, that we illustrate numerically.