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arXiv · 2112.13033

On a skew stable L\'{e}vy process

Alexander Iksanov·Andrey Pilipenko
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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 the noise is a stable L\'{e}vy process with finite mean and infinite variance. We define a skew stable L\'{e}vy process $X$ as the limit of a sequence of stable L\'{e}vy processes which are perturbed at zero. We point out 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 It\^{o}`s excursion theory.

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Alexander Iksanov, Andrey Pilipenko. 2021-12-24. On a skew stable L\'{e}vy process. https://arxiv.org/abs/2112.13033

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