Author:
Cass Thomas,Ferrucci Emilio
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.
Funder
UK Research and Innovation
Publisher
Springer Science and Business Media LLC
Subject
Statistics, Probability and Uncertainty,Statistics and Probability,Analysis