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Regularity of local times associated with Volterra–Lévy processes and path-wise regularization of stochastic differential equations

Harang, Fabian A.; Ling, Chengcheng

We investigate the space-time regularity of the local time associated with Volterra–Lévy processes, including Volterra processes driven by α-stable processes for α∈(0,2]. We show that the spatial regularity of the local time for Volterra–Lévy process is P-a.s. inverse proportional to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbation of ordinary differential equations by a Volterra–Lévy process which has sufficiently regular local time. Following along the lines of Harang and Perkowski (2020), we show existence, uniqueness and differentiability of the flow associated with such equations.
Published in: Journal of Theoretical Probability, 10.1007/s10959-021-01114-4, Springer Nature