Strong solutions of SDE's with rough coefficients

TL;DR

This paper proves the strong existence and regularity of solutions to SDEs driven by Brownian motion with measurable, Markovian drift, using novel techniques from the theory of abstract Wiener spaces.

Contribution

It introduces a new approach leveraging abstract Wiener space techniques to establish strong solutions for SDEs with minimal regularity assumptions.

Findings

Proves strong existence of solutions under weak regularity conditions.

Shows solutions are H-C-regular maps in the sense of Leonard Gross.

Utilizes Girsanov exponential in L^{1+ε} for existence proof.

Abstract

We give a proof of the strong existence and the regularity of stochastic differential equations driven by a Brownian motion and a measurable, Markovian drift without no regularity hypothesis except that the Girsanov exponential associated is in some L^{1+{\epsilon}}(\mu) for some fixed {\epsilon}>0 by using the techniques which are totally novel originating from the abstract Wiener space, in particular the solution is an H-C-regular map in the sense of the theory of Leonard Gross.