Good rough path sequences and applications to anticipating stochastic calculus

TL;DR

This paper investigates anticipative stochastic differential equations driven by rough paths, establishing conditions under which solutions coincide with Stratonovich solutions, and applies these results to Brownian motion for various probabilistic theorems.

Contribution

It introduces a simple condition ensuring rough path solutions are Stratonovich solutions, unifying many anticipative SDE results, especially for Brownian motion.

Findings

Condition satisfied by Brownian motion

Support theorems for anticipative SDEs

Large deviation principles for Brownian-driven SDEs

Abstract

We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path. Neither adaptedness of initial point and vector fields nor commuting conditions between vector field is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. We then show that this condition is satisfied by the Brownian motion. As application, we obtain rather flexible results such as support theorems, large deviation principles and Wong--Zakai approximations for SDEs driven by Brownian motion along anticipating vectorfields. In particular, this unifies many results on anticipative SDEs.