On the Wiener chaos expansion of the signature of a Gaussian process
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Published version
Author(s)
Cass, Thomas
Ferrucci, Emilio
Type
Journal Article
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 ∈ (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.
processes, which contains fractional Brownian motion (fBm) with Hurst parameter
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-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.
Date Issued
2024-08-01
Date Acceptance
2023-12-04
Citation
Probability Theory and Related Fields, 2024, 189, pp.909-947
ISSN
0178-8051
Publisher
Springer
Start Page
909
End Page
947
Journal / Book Title
Probability Theory and Related Fields
Volume
189
Copyright Statement
© The Author(s) 2024
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Identifier
https://link.springer.com/article/10.1007/s00440-023-01255-z
Publication Status
Published
Date Publish Online
2024-01-04