An approximation scheme for SDEs with non-smooth coefficients

TL;DR

This paper develops an approximation scheme for elliptic SDEs with non-smooth coefficients, establishing existence of solution flows and deriving formulas for derivatives of the associated Markov semigroups.

Contribution

It introduces a novel approximation method for elliptic SDEs with non-smooth coefficients, enabling analysis of solution flows and derivative representations.

Findings

Existence of $W_{ ext{loc}}^{1,p}$ solution flows for elliptic SDEs with Hölder continuous coefficients

Development of an approximation scheme for such SDEs

Derivation of a representation for the derivative of the Markov semigroup and an integration by parts formula

Abstract

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 $W_{\loc}^{1,p}$ solution flows for elliptic SDEs with H\"older continuous and $\cap_{p} W_{\loc}^{1,p}$ 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.