Zooming in on a Lévy process at its supremum

Jevgenijs Ivanovs

Published 2016-10-14Version 1

Let $M$ and $\tau$ be the supremum and its time of a L\'evy process $X$ on some finite time interval. It is shown that zooming in on $X$ at its supremum, that is, considering $(a_\eta(X_{\tau+t/\eta}-M))_{t\in\mathbb R}$ as $\eta,a_\eta\rightarrow\infty$, results in $(\xi_t)_{t\in\mathbb R}$ constructed from two independent processes corresponding to some self-similar L\'evy process $S$ conditioned to stay positive and negative. This holds when $X$ is in the domain of attraction of $S$ under the zooming-in procedure as opposed to the classical zooming-out of Lamperti (1962). As an application of this result we provide a limit theorem for the discretization errors in simulation of supremum and its time, which extends the result of Asmussen, Glynn and Pitman (1995) for the Brownian motion. Moreover, a general invariance principle for L\'evy processes conditioned to stay negative is given.