Regularity of Local Times Associated with Volterra–Lévy Processes and Path-Wise Regularization of Stochastic Differential Equations

Fabian A. Harang () and Chengcheng Ling ()
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Fabian A. Harang: University of Oslo
Chengcheng Ling: Bielefeld University

Journal of Theoretical Probability, 2022, vol. 35, issue 3, 1706-1735

Abstract: Abstract We investigate the space-time regularity of the local time associated with Volterra–Lévy processes, including Volterra processes driven by $$\alpha $$ α -stable processes for $$\alpha \in (0,2]$$ α ∈ ( 0 , 2 ] . We show that the spatial regularity of the local time for Volterra–Lévy process is $${\mathbb {P}}$$ 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.

Keywords: Stochastic differential equations; Lévy process; Volterra process; Regularization by noise; Occupation measure; Local time; Young integral; Stochastic Sewing Lemma; Primary 60H10; 35R09; Secondary 60G51 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01114-4

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