A Wong-Zakai theorem for SDEs with singular drift

Abstract

We study stochastic differential equations (SDEs) with multiplicative Stratonovich-type noise of the form $d{X}_t=b(X_t)dt+\sigma (X_t)\circ d{W}_t,X_0=x_0\in {\mathbb{R}}^d,t\ge 0,$ with a possibly singular drift $b\in L^p({\mathbb{R}}^d)$, $p>d$ and $p\ge 2$, 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.

19 pages