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

Abstract

AbstractWe compute the Wiener chaos decomposition of the signature for a class of Gaussian processes, which contains fractional Brownian motion (fBm) with Hurst parameter $$H \in (1/4,1)$$ H ∈ ( 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 $$1/2 < H$$ 1 / 2 < H -fBm (Baudoin and Coutin in Stochast Process Appl 117(5):550–574, 2007) and the one for Brownian motion ($$H = 1/2$$ H = 1 / 2 ) (Fawcett 2003), to the general case $$H > 1/4$$ H > 1 / 4 , thereby resolving an established open problem. Other processes studied include continuous and centred Gaussian semimartingales.