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

Emilio Ferrucci; Thomas Cass

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arXiv · 2207.08422

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

Emilio Ferrucci·Thomas Cass

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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 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.

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Emilio Ferrucci, Thomas Cass. 2022-07-18. On the Wiener Chaos Expansion of the Signature of a Gaussian Process. https://arxiv.org/abs/2207.08422

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