Bridging the first and last passage times for Lévy models

Abstract

Research in classical ruin theory has largely focused on the first passage time analysis of a surplus process below level 0. Recently, inspired by numerous applications in finance, physics, and optimization, there has been an accrued interest in the analysis of the last passage time (below level 0). In this paper, we aim to bridge the first and the last passage times and unify their analyses. For this purpose, we consider negative excursions of an underlying process in two manners, cumulative and noncumulative, and introduce two random times, denoted by s _r and l _r, where r can be interpreted as a measure of a decision maker's tolerance to negative excursions. Our analysis focuses on spectrally negative Lévy processes, for which we derive the Laplace transform and some distributional quantities of these random times in terms of standard scale functions. An application to credit risk management is considered at the end.

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