2021/12/24 by Alexander Iksanov, Iksanov, Alexander, Andrey Pilipenko +1
Economics, Econometrics and Finance · Mathematics · #60J35 #FOS: Mathematics #Financial Risk and Volatility Modeling #Primary: 60F17 #Probability (math.PR) #Secondary: 60J50 #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2112.13033
openalex publication_date 2021/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Lévy process with finite mean and infinite variance. We define a skew stable Lévy process X as the limit of a sequence of stable Lévy 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 Itô`s excursion theory.