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November 2020 Signature cumulants, ordered partitions, and independence of stochastic processes
Patric Bonnier, Harald Oberhauser
Bernoulli 26(4): 2727-2757 (November 2020). DOI: 10.3150/20-BEJ1205

Abstract

The sequence of so-called signature moments describes the laws of many stochastic processes in analogy with how the sequence of moments describes the laws of vector-valued random variables. However, even for vector-valued random variables, the sequence of cumulants is much better suited for many tasks than the sequence of moments. This motivates us to study so-called signature cumulants. To do so, we develop an elementary combinatorial approach and show that in the same way that cumulants relate to the lattice of partitions, signature cumulants relate to the lattice of so-called “ordered partitions”. We use this to give a new characterisation of independence of multivariate stochastic processes. Finally, we construct a family of unbiased minimum-variance estimators of signature cumulants and show that even for the simple example of a diffusion with constant drift and volatility, such signature cumulant estimators outperform signature moment estimators.

Citation

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Patric Bonnier. Harald Oberhauser. "Signature cumulants, ordered partitions, and independence of stochastic processes." Bernoulli 26 (4) 2727 - 2757, November 2020. https://doi.org/10.3150/20-BEJ1205

Information

Received: 1 September 2019; Revised: 1 January 2020; Published: November 2020
First available in Project Euclid: 27 August 2020

zbMATH: 07256158
MathSciNet: MR4140527
Digital Object Identifier: 10.3150/20-BEJ1205

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 4 • November 2020
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