Large Deviations for the Maximum Local Time of Stable Levy Processes

Abstract

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.