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May 2021 Time-changed spectrally positive Lévy processes started from infinity
Clément Foucart, Pei-Sen Li, Xiaowen Zhou
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Bernoulli 27(2): 1291-1318 (May 2021). DOI: 10.3150/20-BEJ1274

Abstract

Consider a spectrally positive Lévy process Z with log-Laplace exponent Ψ and a positive continuous function R on (0,). We investigate the entrance from infinity of the process X obtained by changing time in Z with the inverse of the additive functional η(t)=0tdsR(Zs). We provide a necessary and sufficient condition for infinity to be an entrance boundary of the process X. Under this condition, the process can start from infinity and we study its speed of coming down from infinity. When the Lévy process has a negative drift δ:=γ<0, sufficient conditions over R and Ψ are found for the process to come down from infinity along the deterministic function (xt,t0) solution to dxt=γR(xt)dt with x0=. If Ψ(λ)cλα as λ0, α(1,2], c>0 and R is regularly varying at with index θ>α, the process comes down from infinity and we find a renormalisation in law of its running infimum at small times.

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Clément Foucart. Pei-Sen Li. Xiaowen Zhou. "Time-changed spectrally positive Lévy processes started from infinity." Bernoulli 27 (2) 1291 - 1318, May 2021. https://doi.org/10.3150/20-BEJ1274

Information

Received: 1 January 2020; Revised: 1 May 2020; Published: May 2021
First available in Project Euclid: 24 March 2021

Digital Object Identifier: 10.3150/20-BEJ1274

Keywords: coming down from infinity , continuous-state non-linear branching process , entrance boundary , hitting time , regularly varying function , spectrally positive Lévy process , time-change , weighted occupation time

Rights: Copyright © 2021 ISI/BS

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Vol.27 • No. 2 • May 2021
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