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

Abstract

This paper is concerned with the finiteness and large-time behaviour of moments of the overshoot and undershoot of a high level, and of their moment generating functions (MGFs), for a Lévy process which drifts to -∞ almost surely. 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.

Computing the Ruin Probability of Lévy Insurance Risk Processes in non-Cramér Models

“…At this stage, we bring in the convolution equivalent assumptions. We assume the same conditions as in Park and Maller (2008) Throughout the entire paper the relationship of the above assumption is mentioned well in Section 4 of Klüppelberg et al (2004). Assume that the process is not spectrally negative, in fact, assume Π X {(x, ∞)} > 0 for all x > 0.…”

“…Park and Maller (2008) and Klüppelberg et al (2004) introduced the limiting probability when causes the ruin as follows:…”

“…Specific illustrations of the theoretical results are given for special cases such as VG and NIG models. In this analysis we will follow the notation of Park and Maller (2008).…”

Stability of the exit time for Lévy processes

“…Applications of this and related kinds of result abound; we have in mind, in particular, applications to the insurance risk process; see, e.g. recent results in [5], [15], [19], [28], and [31]. These authors have tended to concentrate on properties of the overshoot and undershoots, with less attention paid to the ruin time, τ u .…”

Asymptotics for the moments of the overshoot and undershoot of a random walk

“…The results of this paper can also be applied to other fields of applied probability. Recently, Park and Maller (2008) considered the nonlocal asymptotics of the β-order moment of the overshoot and undershoot of a Lévy process with drifts to −∞ almost surely (here β is a positive constant). Since a Lévy process is different from a random walk, in forthcoming papers we will investigate the uniform local asymptotics of a Lévy process and its overshoot and undershoot, and the local asymptotics of the ϕ-moments of the overshoot and undershoot of a Lévy process.…”