On the Wiener Chaos Expansion of the Signature of a Gaussian Process

TL;DR

This paper derives the Wiener chaos decomposition of the signature for certain Gaussian processes, including fractional Brownian motion with Hurst parameter H in (1/4, 1), extending previous results and solving an open problem.

Contribution

It provides a unified Wiener chaos expansion for the signature of Gaussian processes, generalizing known formulas for Brownian motion and fractional Brownian motion.

Findings

Derived explicit Wiener chaos decomposition for Gaussian process signatures

Extended formulas to fractional Brownian motion with H in (1/4, 1)

Resolved an open problem in the field

Abstract

We compute the Wiener chaos decomposition of the signature for a class of Gaussian processes, which contains fractional Brownian motion (fBm) with Hurst parameter H in (1/4, 1). At level 0, our result yields an expression for the expected signature of such processes, which determines their law [CL16]. In particular, this formula simultaneously extends both the one for 1/2 < H-fBm [BC07] and the one for Brownian motion (H = 1/2) [Faw03], to the general case H > 1/4, thereby resolving an established open problem. Other processes studied include continuous and centred Gaussian semimartingales.