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

2014/10/02 by Torstein Nilssen, Tusheng Zhang, Nilssen, Torstein +1
Economics, Econometrics and Finance · Engineering · Mathematics · #60H07 #60H10 #60H30 #FOS: Mathematics #Navier-Stokes equation solutions #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications #math.PR #msc:60H07 #msc:60H10 #msc:60H30

paper · pdf · doi:10.48550/arxiv.1410.0520

arxiv created 2014/10/02 · openalex publication_date 2014/10/02 · arxiv updated 2014/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• On the strong solutions of one-dimensional stochastic differential equations with reflecting boundary

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