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2020 Compactness and continuity properties for a Lévy process at a two-sided exit time
Ross A. Maller, David M. Mason
Electron. J. Probab. 25: 1-26 (2020). DOI: 10.1214/20-EJP451

Abstract

We consider a Lévy process $X=(X(t))_{t\ge 0}$ in a generalised Feller class at 0, and study the exit position, $\left \vert X(T(r))\right \vert $, as $X$ leaves, and the position, $\left \vert X(T( r) -)\right \vert $, just prior to its leaving, at time $T(r)$, a two-sided region with boundaries at $\pm r$, $r>0$. Conditions are known for $X$ to be in the Feller class $FC_{0}$ at zero, by which we mean that each sequence $t_{k}\downarrow 0$ contains a subsequence through which $X(t_{k})$, after norming by a nonstochastic function, converges to an a.s. finite nondegenerate random variable. We use these conditions on $X$ to characterise similar properties for the normed positions $\left \vert X(T( r))\right \vert /r$ and $\left \vert X(T( r) -)\right \vert /r$, and also for the normed jump $\left \vert \Delta X(T(r))/r\right \vert = \left \vert X(T(r))-X(T(r)-)\right \vert /r$ (“the jump causing ruin"), as convergence takes place through sequences $r_{k}\downarrow 0$. We go on to give conditions for the continuity of distributions of the limiting random variables obtained in this way.

Version Information

This article was first posted online with the misprints in the second to last paragraph in section 1.1. Misprints have been corrected on 19 May 2020.

Citation

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Ross A. Maller. David M. Mason. "Compactness and continuity properties for a Lévy process at a two-sided exit time." Electron. J. Probab. 25 1 - 26, 2020. https://doi.org/10.1214/20-EJP451

Information

Received: 26 July 2019; Accepted: 7 April 2020; Published: 2020
First available in Project Euclid: 29 April 2020

zbMATH: 1440.62066
Digital Object Identifier: 10.1214/20-EJP451

Subjects:
Primary: 62B15, 62E17, 62G05

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