A Wong-Zakai theorem for SDEs with singular drift

TL;DR

This paper establishes a Wong-Zakai type approximation for SDEs with singular drifts and multiplicative noise, showing they can be approximated by smoothed ODEs and providing a support theorem for such equations.

Contribution

It introduces a novel approximation method for SDEs with singular drifts using simultaneous smoothing of noise and drift, and proves a support theorem for this class.

Findings

SDEs with singular drifts can be approximated by smoothed ODEs.

A support theorem is established for these SDEs using Girsanov's theorem.

The approach simplifies the analysis of such SDEs with irregular coefficients.

Abstract

We study stochastic differential equations (SDEs) with multiplicative Stratonovich-type noise of the form $ dX_t = b(X_t) dt + \sigma(X_t)\circ d W_t, X_0=x_0\in\mathbb{R}^d, t\geq0,$ with a possibly singular drift $b\in L^{{p}}(\mathbb{R}^d)$, $p>d$ and $p\geq 2$, and show that such SDEs can be approximated by random ordinary differential equations by smoothing the noise and the singular drift at the same time. We further prove a support theorem for this class of SDEs in a rather simple way using the Girsanov theorem.