Numerical methods for SDEs with drift discontinuous on a set of positive reach

TL;DR

This paper develops a strongly convergent numerical scheme for SDEs with drift coefficients that are piecewise Lipschitz and have discontinuities on smooth hypersurfaces of positive reach, extending classical results beyond Lipschitz conditions.

Contribution

It introduces a new class of SDEs with piecewise Lipschitz drifts and constructs a convergent numerical method under geometric conditions on discontinuities.

Findings

Existence of solutions under weaker regularity conditions.

Construction of a strongly convergent approximation scheme.

Applicability to SDEs with discontinuous drift on smooth hypersurfaces.

Abstract

For time-homogeneous stochastic differential equations (SDEs) it is enough to know that the coefficients are Lipschitz to conclude existence and uniqueness of a solution, as well as the existence of a strongly convergent numerical method for its approximation. Here we introduce a notion of piecewise Lipschitz functions and study SDEs with a drift coefficient satisfying only this weaker regularity condition. For these SDEs we can construct a strongly convergent approximation scheme, if the set of discontinuities is a sufficiently smooth hypersurface satisfying the geometrical property of being of positive reach. We then arrive at similar conclusions as in the Lipschitz case. We will see that, although SDEs are in the center of our interest, we will talk surprisingly little about probability theory here.