Overshoots and undershoots of Lévy processes

Abstract

We obtain a new fluctuation identity for a general Lévy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Klüppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766-1801] concerning asymptotic overshoot distribution of a particular class of Lévy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying Lévy process is spectrally one sided.

Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem

“…At the same time, there has been a growing body of literature concerning actuarial mathematics which explores the interaction of classical models of risk and fine properties of Lévy processes with a view to gaining new results on both sides (see for example [2,10,18,19,23,22,24,26,28,31]). …”

Moment and MGF convergence of overshoots and undershoots for Lévy insurance risk processes

“…This provides information relevant to quantities associated with the ruin of an insurance risk process. Results of Klüppelberg, Kyprianou, and Maller (2004) and Doney and Kyprianou (2006) for asymptotic overshoot and undershoot distributions in the class of Lévy processes with convolution equivalent canonical measures are shown to have moment and MGF convergence extensions. …”

“…Thus, (H t ) t≥0 will define the ascending ladder height subordinator of X, which is defective under the assumption that lim t→∞ X t = −∞ a.s. (which we will always assume in this paper), taking the value +∞ once X exceeds its all-time maximum. The (proper) descending ladder height subordinator is denoted by (Ĥ t ) t≥0 (taking positive values; we depart from the notation of [9] here, in favour of the setup in Doney and Kyprianou [3]). …”

A Note on Scale Functions and the Time Value of Ruin for Lévy Insurance Risk Processes

“…According to the quintuple law (see Example 8 in Doney and Kyprianou, 2006), in the case that X drifts to ∞ (and hence Φ(0) = 0), we have for u, v > 0 and 0 < y ≤ v ∧ x,…”

“…Here one also finds a growing body of work (e.g. : Doney and Kyprianou, 2006;Klüppelberg and Erder, 2008;Breuer, 2008;Kyprianou, Pardo and Rivero, 2008a;Chaumont, Kyprianou and Pardo, 2009;Caballero and Chaumont, 2006;Chen and Sheu, 2009).…”