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

Philip S. Griffin, Dale O. Roberts

Abstract: 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.

Journal: Stochastic Processes and their Applications

DOI: 10.1016/j.spa.2015.11.005