Recent models of the insurance risk process use a Levy process to generalise the traditional Cramer-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a Levy process which drifts to -infinity and satis es a Cramer or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cramer case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parameters tend to equality. Thus, at least regarding these quantities, the "medium-heavy" tailed convolution equivalent model segues into the "light-tailed" Cramer model in a natural way. This suggests a usefully expanded exibility for modelling the insurance risk process. We illustrate this relationship by
comparing the asymptotic distributions obtained for the overshoot and undershoots, assuming the Levy process belongs to the "GTSC" class.
Griffin, Philip S.; Maller, Ross A.; and van Schaik, Kees, "Asymptotic Distributions of the Overshoot and Undershoots for the Lévy Insurance Risk Process in the Cramér and Convolution Equivalent Cases" (2011). Mathematics Faculty Scholarship. Paper 94.
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