The Moments of L\'evy's area using a sticky shuffle Hopf algebra

TL;DR

This paper presents a combinatorial method using a sticky shuffle Hopf algebra to evaluate the moments of Lévy's stochastic area, simplifying previous approaches and confirming their results.

Contribution

It introduces a new combinatorial approach with Hopf algebra techniques to compute Lévy's area moments, simplifying existing methods.

Findings

Lévy's area moments are expressed via Euler numbers.

The combinatorial approach simplifies previous calculations.

The method confirms recent results by Levin and Wildon.

Abstract

L\'evy's stochastic area for planar Brownian motion is the difference of two iterated integrals of second rank against its component one-dimen\-sional Brownian motions. Such iterated integrals can be multiplied using the sticky shuffle product determined by the underlying It\^o algebra of stochastic differentials. We use combinatorial enumerations that arise from the distributive law in the corresponding Hopf algebra structure to evaluate the moments of L\'evy's area. These L\'evy moments are well known to be given essentially by the Euler numbers. This has recently been confirmed in a novel combinatorial approach by Levin and Wildon. Our combinatorial calculations considerably simplify their approach.