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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


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.

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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.


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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.


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

Primary: 62B15 , 62E17 , 62G05

Keywords: domain of partial attraction , generalised Feller class , Lévy process , passage time distributions , stochastic compactness , two-sided exit problem


Vol.25 • 2020
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