Regularity of Local times associated to Volterra-L\'evy processes and path-wise regularization of stochastic differential equations

TL;DR

This paper studies the regularity of local times for Volterra-Lévy processes, revealing how the kernel's singularity affects regularity and applying these findings to path-wise regularization of stochastic differential equations.

Contribution

It provides new insights into the space-time regularity of local times for Volterra-Lévy processes and applies these results to establish existence and uniqueness of solutions for perturbed differential equations.

Findings

Local time regularity inversely related to kernel singularity

Path-wise regularization effects for differential equations with Volterra-Lévy noise

Existence and differentiability of solution flows

Abstract

We investigate the space-time regularity of the local time associated to Volterra-L\'evy processes, including Volterra processes driven by $\alpha$-stable processes for $\alpha\in(0,2]$. We show that the spatial regularity of the local time for Volterra-L\'evy process is $P$-a.s. inverse proportionally to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturba\Ption of ODEs by a Volterra-L\'evy process which has sufficiently regular local time. Following along the lines of [15], we show existence, uniqueness and differentiablility of the flow associated to such equations.