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Zooming in on a Lévy process at its supremum

By Jevgenijs Ivanovs
Thiele Research Reports
No. 05, November 2016

Let $M$ and $\tau$ be the supremum and its time of a Lévy process $X$ on some finite time interval. It is shown that zooming in on $X$ at its supremum, that is, considering $((X_{\tau+t\varepsilon}-M)/a_\varepsilon)_{t\in\mathbb{R}}$ as $\varepsilon\downarrow 0$, results in $(\xi_t)_{t\in\mathbb{R}}$ constructed from two independent processes having the laws of some self-similar Lévy process $\widehat{X}$ conditioned to stay positive and negative. This holds when $X$ is in the domain of attraction of $\widehat{X}$ under the zooming-in procedure as opposed to the classical zooming out of Lamperti (1962). As an application of this result we establish 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. Additionally, complete characterization of the domains of attraction when zooming in on a Lévy process at 0 is provided.

Keywords: conditioned to stay positive, discretization error, domains of attraction, Euler scheme, functional limit theorem, high frequency statistics, invariance principle, scaling limits, self-similarity, small-time behaviour

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