The moments of Lévy's area using a sticky shuffle Hopf algebra

Lévy's stochastic area for planar Brownian motion is the dif- ference of two iterated integrals of second rank against its component one- dimensional Brownian motions. Such iterated integrals can be multiplied using the sticky shuffle product determined by the underlying Itô 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 Lévy's area. These Lévy 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.