A Note on the Malliavin Differentiability of One-Dimensional Reflected Stochastic Differential Equations with Discontinuous Drift

TL;DR

This paper extends the understanding of Malliavin differentiability to one-dimensional reflected SDEs with discontinuous, bounded drift, and applies these results to derive a Bismut-Elworthy-Li formula for the associated Kolmogorov equation.

Contribution

It demonstrates that reflected SDEs with discontinuous drift are Malliavin differentiable, similar to non-reflected cases, and derives a related Bismut-Elworthy-Li formula.

Findings

Reflected SDEs with discontinuous drift are Malliavin differentiable.

Established a Bismut-Elworthy-Li formula for the Kolmogorov equation.

Extended regularity results to reflected stochastic differential equations.

Abstract

We consider a one-dimensional Stochastic Differential Equation with reflection where we allow the drift to be merely bounded and measurable. It is already known that such equations have a unique strong solution. Recently, it has been shown that non-reflected SDE's with discontinuous drift possess more regularity than one could expect, namely they are Malliavin differentiable and weakly differentiable w.r.t. the initial value. In this paper we show that similar results hold for one-dimensional SDE's with reflection. We then apply the results to get a Bismut-Elworthy-Li formula for the corresponding Kolmogorov equation.