Approximations of the Brownian Rough Path with Applications to Stochastic Analysis

Abstract

A geometric p-rough path can be seen to be a genuine path of finite p-variation with values in a Lie group equipped with a natural distance. The group and its distance lift (R^{d},+,0) and its Euclidean distance. This approach allows us to easily get a precise modulus of continuity for the Enhanced Brownian Motion (the Brownian Motion and its Levy Area). As a first application, extending an idea due to Millet & Sanz-Sole, we characterize the support of the Enhanced Brownian Motion (without relying on correlation inequalities). Secondly, we prove Schilder's theorem for this Enhanced Brownian Motion. As all results apply in Hoelder (and stronger) topologies, this extends recent work by Ledoux, Qian, Zhang (2002). Lyons' fine estimates in terms of control functions allow us to show that the Ito map is still continuous in the topologies we introduced. This provides new and simplified proofs of the Stroock-Varadhan support theorem and the Freidlin-Wentzell theory. It also provides a short proof of modulus of continuity for diffusion processes along old results by Baldi.