The Moments of Lévy's Area Using a Sticky Shuffle Hopf Algebra

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.