An approximation scheme for SDEs with non-smooth coefficients

Elliptic stochastic differential equations (SDE) make sense when the coefficients are only continuous. We study the corresponding linearized SDE whose coefficients are not assumed to be locally bounded. This leads to existence of $W_{\text{loc}}^{1,p}$ solution flows for elliptic SDEs with H"older continuous and ${\cap }_p{W}_{\text{loc}}^{1,p}$ coefficients. Furthermore an approximation scheme is studied from which we obtain a representation for the derivative of the Markov semigroup, and an integration by parts formula.