Lévy's stochastic area revisited using a sticky shuffle Hopf algebra

07.12.2016, 10:45  –  Haus 9, Raum 0.13
Arbeitsgruppenseminar Analysis

Yue Wu (TU Berlin)

Lévy's area is simply the signed area enclosed by the planar
Brownian motion and its chord. It is intrinsically the difference of two
iterated integrals of second rank against its component one-dimensional
Brownian motions. This sort of stochastic iterated integrals can be
multiplied using the sticky shuffle product determined by the underlying
Itô algebra of stochastic differentials. We then 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.