On properties of a flow generated by an SDE with discontinuous drift

TL;DR

This paper investigates the differentiability and asymptotic behavior of stochastic flows on the real line generated by SDEs with discontinuous, bounded variation drifts, providing insights into their mathematical properties.

Contribution

It establishes the differentiability of flows generated by SDEs with discontinuous drifts and analyzes their asymptotic properties, which was previously not well understood.

Findings

Flow is differentiable with respect to initial conditions.

Asymptotic properties of the flow are characterized.

Results extend understanding of SDEs with discontinuous drifts.

Abstract

We consider a stochastic flow on $\mathds{R}$ generated by an SDE with its drift being a function of bounded variation. We show that the flow is differentiable with respect to the initial conditions. Asymptotic properties of the flow are studied.