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

2022/07/18 by Emilio Rossi Ferrucci, Ferrucci, Emilio, Thomas Cass +1
Economics, Econometrics and Finance · #60H07 #60L10 #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2207.08422

openalex publication_date 2022/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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