A representation for the Expected Signature of Brownian motion up to the first exit time of the planar unit disc

Abstract

The signature of a sample path is a formal series of iterated integrals along the path. The expected signature of a stochastic process gives a summary of the process that is especially useful for studying stochastic differential equations driven by the process. Lyons-Ni derived a partial differential equation for the expected signature of Brownian motion, starting at a point z in a bounded domain, until it hits to boundary of the domain. We focus on the domain of planar unit disc centred at 0. Motivated by recently found explicit formulae for some terms in the expected signature of this process in terms of Bessel functions, we derive a tensor series representation for this expected signature, coming from from studying Lyons-Ni's PDE. Although the representation is rather involved, it simplifies significantly to give a formula for the polynomial leading order term in each tensor component of the expected signature.