On a skew stable Lévy process

Alexander Iksanov and Andrey Pilipenko

Stochastic Processes and their Applications, 2023, vol. 156, issue C, 44-68

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.

Keywords: Excursion theory; Functional limit theorem; Recurrent extension of a Markov process; Skew Brownian motion; Stable Lévy process; Stochastic differential equation with a local time (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2022.11.004

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