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

Author

Listed:
  • Fabian A. Harang

    (University of Oslo)

  • Chengcheng Ling

    (Bielefeld University)

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.

Suggested Citation

  • Fabian A. Harang & Chengcheng Ling, 2022. "Regularity of Local Times Associated with Volterra–Lévy Processes and Path-Wise Regularization of Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1706-1735, September.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:3:d:10.1007_s10959-021-01114-4
    DOI: 10.1007/s10959-021-01114-4
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    References listed on IDEAS

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    1. Rosinski, Jan, 1986. "On stochastic integral representation of stable processes with sample paths in Banach spaces," Journal of Multivariate Analysis, Elsevier, vol. 20(2), pages 277-302, December.
    2. Catellier, R. & Gubinelli, M., 2016. "Averaging along irregular curves and regularisation of ODEs," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2323-2366.
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