Rough path theory and applications to stochastic analysis.

摘要

Rough Path Theory á la Terry Lyons is a purely deterministic theory of differential equations driven by signals of very little regularity. In particular, it applies to Stochastic Differential Equations driven by Brownian Motion and allows powerful insights into classical diffusion theory.; In essence; Brownian Motion and Lévy-area (Enhanced Brownian Motion) are identified as correct driving signals of SDEs. In addition. Lyons proved a continuity statement (Universal, Limit Theorem) in a p-variation-type topology.; We find that a more natural Hölder-continuity follows from fine estimates in his work. A number of approximations to EBM are studied and they lead to a new straightforward proof of the classical Stroock-Varadhan Support Theorem for diffusions.; We also consider Large Deviations for EBM and construct exponentially good approximations based on Subriemannian Geodesics.