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On a skew stable Lévy process

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  • Iksanov, Alexander
  • Pilipenko, Andrey

Abstract

The skew Brownian motion is a strong Markov process which behaves like a Brownian motion until hitting zero and exhibits an asymmetry at zero. We address the following question: what is a natural counterpart of the skew Brownian motion in the situation that an underlying Brownian motion is replaced with a stable Lévy process with finite mean and infinite variance. We define a skew stable Lévy process X as a limit of a sequence of stable Lévy processes which are perturbed at zero. We derive a formula for the resolvent of X and show that X is a solution to a stochastic differential equation with a local time. Also, we provide a representation of X in terms of Itô‘s excursion theory.

Suggested Citation

  • Iksanov, Alexander & Pilipenko, Andrey, 2023. "On a skew stable Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 44-68.
    • Handle: RePEc:eee:spapps:v:156:y:2023:i:c:p:44-68
    DOI: 10.1016/j.spa.2022.11.004
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    References listed on IDEAS

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    1. David Baños & Salvador Ortiz-Latorre & Andrey Pilipenko & Frank Proske, 2022. "Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise," Journal of Theoretical Probability, Springer, vol. 35(2), pages 714-771, June.
    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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