Convolution equivalent Lévy processes and first passage times

Abstract

We investigate the behavior of Lévy processes with convolution equivalent Lévy measures, up to the time of first passage over a high level u. Such problems arise naturally in the context of insurance risk where u is the initial reserve. We obtain a precise asymptotic estimate on the probability of first passage occurring by time T. This result is then used to study the process conditioned on first passage by time T. The existence of a limiting process as $u\to \infty$ is demonstrated, which leads to precise estimates for the probability of other events relating to first passage, such as the overshoot. A discussion of these results, as they relate to insurance risk, is also given.

Published in at http://dx.doi.org/10.1214/12-AAP879 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Ruin Probabilities and Overshoots for General Levy Insurance Risk Processes
• Overshoots and undershoots of Lévy processes
• Path decomposition of ruinous behavior for a general Lévy insurance risk process
• Asymptotic Distributions of the Overshoot and Undershoots for the Lévy Insurance Risk Process in the Cramér and Convolution Equivalent Cases

Cited by in corpus (4)

• Interplay of insurance and financial risks in a discrete-time model with strongly regular variation
• Sample path behavior of a Lévy insurance risk process approaching ruin, under the Cramér-Lundberg and convolution equivalent conditions
• On the gamma-reflected processes with fBm input
• Gaussian risk models with financial constraints