Modern Stochastics: Theory and Applications

A novel theoretical result on estimation of the local time and the occupation time measure of an α-stable Lévy process with $\alpha \in (1,2)$ is presented. The approach is based upon computing the conditional expectation of the desired quantities given high frequency data, which is an ${L^{2}}$-optimal statistic by construction. The corresponding stable central limit theorems are proved and a statistical application is discussed. In particular, this work extends the results of [20], which investigated the case of the Brownian motion.

We prove a large deviation type estimate for the asymptotic behavior of a weighted local time of $\varepsilon W$ as $\varepsilon \to 0$.