Expected signature of two dimensional Brownian Motion up to the first exit time of the domain

Details Expected signature of two dimensional Brownian Motion up to the first exit time of the domain Expected signature of two dimensional Brownian Motion up to the first exit time of the domain

2011-01-31

arxiv.org/abs/1101.5902v4

Mathematics

Probability

Scientific paper

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 paper 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 = R2. 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. It is true for a higher dimensional Brownian Motion as well. We are interested in generalizing our results to embedded Markovian diffussion processes.

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