On the Wiener chaos expansion of the signature of a Gaussian process

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 (Chevyrev and Lyons in Ann Probab 44(6):4049–4082, 2016). In particular, this formula simultaneously extends both the one for \(1/2 < H\)-fBm (Baudoin and Coutin in Stochast Process Appl 117(5):550–574, 2007) and the one for Brownian motion (\(H = 1/2\)) (Fawcett 2003), to the general case \(H > 1/4\), thereby resolving an established open problem. Other processes studied include continuous and centred Gaussian semimartingales.