Let $X(t)$ be a strictly stable Levy process of index $1 < \alpha \leq 2$ and skewness index $|h| \leq 1$. Let $L^x_t$ be its local time and $L^\ast_t = \sup_x L^x_t$ the maximum local time. We show that $\lim_{\lambda \rightarrow + \infty} \lambda^{-\alpha} \log P(L^\ast_1 > \lambda) = -C_{\alpha h}$, where $C_{\alpha h}$ is a known constant. In the case that $X(t)$ is a standard Brownian motion, $C_{\alpha h} = 1/2$ and the result is due to Perkins.
Publié le : 1990-10-14
Classification:
Maximum local time,
stable processes,
large deviations,
60J55,
60E15,
60G17
@article{1176990640,
author = {Lacey, Michael},
title = {Large Deviations for the Maximum Local Time of Stable Levy Processes},
journal = {Ann. Probab.},
volume = {18},
number = {4},
year = {1990},
pages = { 1669-1675},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990640}
}
Lacey, Michael. Large Deviations for the Maximum Local Time of Stable Levy Processes. Ann. Probab., Tome 18 (1990) no. 4, pp. 1669-1675. http://gdmltest.u-ga.fr/item/1176990640/