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

Understand

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.