Sample paths of a Lévy process leading to first passage over high levels in finite time

Philip S. Griffin; Dale O. Roberts

doi:10.1016/j.spa.2015.11.005

Sample paths of a Lévy process leading to first passage over high levels in finite time

Philip S. Griffin, Dale O. Roberts^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

Let X be a Lévy process and τ(u)=inf{t:X_t>u} the first passage time of X over level u. For fixed T<∞, sharp asymptotic estimates for P(τ(u)<T) as u→∞ have been developed for several classes of Lévy processes. In this paper we investigate the asymptotic behavior of the sample paths of the process which lead to first passage by time T. This complements previous work in the T=∞ case and is motivated, in part, by problems in insurance risk.

Original language	English
Pages (from-to)	1331-1352
Number of pages	22
Journal	Stochastic Processes and their Applications
Volume	126
Issue number	5
DOIs	https://doi.org/10.1016/j.spa.2015.11.005
Publication status	Published - May 2016

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abstract = "Let X be a L{\'e}vy process and τ(u)=inf{t:Xt>u} the first passage time of X over level u. For fixed T<∞, sharp asymptotic estimates for P(τ(u)<T) as u→∞ have been developed for several classes of L{\'e}vy processes. In this paper we investigate the asymptotic behavior of the sample paths of the process which lead to first passage by time T. This complements previous work in the T=∞ case and is motivated, in part, by problems in insurance risk.",

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AU - Griffin, Philip S.

AU - Roberts, Dale O.

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AB - Let X be a Lévy process and τ(u)=inf{t:Xt>u} the first passage time of X over level u. For fixed T<∞, sharp asymptotic estimates for P(τ(u)<T) as u→∞ have been developed for several classes of Lévy processes. In this paper we investigate the asymptotic behavior of the sample paths of the process which lead to first passage by time T. This complements previous work in the T=∞ case and is motivated, in part, by problems in insurance risk.

KW - Asymptotic

KW - Convolution equivalent

KW - First passage within finite time

KW - Insurance risk

KW - Lévy processes

KW - Semi-heavy tails

KW - Subexponential

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DO - 10.1016/j.spa.2015.11.005

M3 - Article

SN - 0304-4149

VL - 126

SP - 1331

EP - 1352

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 5

ER -

Sample paths of a Lévy process leading to first passage over high levels in finite time

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