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Zooming in on a lévy process at its supremum

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Let M and τ 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 ((Xτ+tε − M)/aε)tR as ε ↓ 0, results in (ξt)tR constructed from two independent processes having the laws of some self-similar Lévy process X conditioned to stay positive and negative. This holds when X is in the domain of attraction of X under the zooming-in procedure as opposed to the classical zooming out [Trans. Amer. Math. Soc. 104 (1962) 62–78]. 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 in [Ann. Appl. Probab. 5 (1995) 875–896] for a linear Brownian motion. Additionally, complete characterization of the domains of attraction when zooming in on a Lévy process is provided.

Original languageEnglish
JournalAnnals of Applied Probability
Pages (from-to)912-940
Number of pages29
Publication statusPublished - 1 Apr 2018

    Research areas

  • Conditioned to stay positive, Discretization error, Domains of attraction, Euler scheme, Functional limit theorem, High frequency statistics, Invariance principle, Mixing convergence, Scaling limits, Self-similarity, Small-time behaviour.

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