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

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• 作者：Philip S. Griffin^a ; ^{psgriffi@syr.edu" class="auth_mail" title="E-mail the corresponding author} ; Dale O. Roberts^b ; ^{dale.roberts@anu.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60G51 ; 60F17 ; 91B30 ; 62P05
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：126
• 期：5
• 页码：1331-1352
• 全文大小：288 K

文摘

Let b4008834b8b" title="Click to view the MathML source">X$X$ be a Lévy process and τ(u)=inf{t:X_t>u}$\tau \left(u\right)=inf\{t:X_t>u\}$ the first passage time of b4008834b8b" title="Click to view the MathML source">X$X$ over level u$u$. For fixed T<∞$T<\infty$, sharp asymptotic estimates for P(τ(u)<T)$P(\tau \left(u\right)<T)$ as u→∞$u\to \infty$ 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$T$. This complements previous work in the T=∞$T=\infty$ case and is motivated, in part, by problems in insurance risk.