The expected signature of brownian motion upon the first exit time of a regular domain

The signature of the path is an essential object in rough path theory which takes value in tensor algebra and it is anticipated that the expected signature of Brownian motion might characterize the rough path measure of Brownian path itself. In this presentation we study the expected signature of a Brownian path in a Bananch space E stopped at the first exit time of an arbitrary regular domain, although we will focus on the case E=R^{2}. We prove that such expected signature of Brownian motion should satisfy one particular PDE and using the PDE for the expected signature and the boundary condition we can derive each term of expected signature recursively. We expect our method to be generalized to higher dimensional case in R^{d}, where d is an integer and d >= 2.