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

• 成果类型: Article
• 署名作者: Cass, Thomas; Ferrucci, Emilio
• 署名单位: Imperial College London; University of Oxford
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01255-z
• 发表日期: 2024
• 页码: 909-947
• 关键词: rates; CONVERGENCE; regression

摘要

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 is an element of (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.