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Asymptotic Distributions of the Overshoot and Undershoots for the L'evy Insurance Risk Process in the Cram'er and Convolution Equivalent Cases | | Philip S Griffin
; Ross A Maller
; Kees van Schaik
; | | Date: |
16 Jun 2011 | | Abstract: | Recent models of the insurance risk process use a L’evy process to
generalise the traditional Cram’er-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 L’{e}vy process which drifts to $-infty$
and satisfies a Cram’er or a convolution equivalent condition. We derive these
asymptotics under minimal conditions in the Cram’er 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" Cram’er model in a
natural way. This suggests a usefully expanded flexibility for modelling the
insurance risk process. We illustrate this relationship by comparing the
asymptotic distributions obtained for the overshoot and undershoots, assuming
the L’evy process belongs to the "GTSC" class. | | Source: | arXiv, 1106.3292 | | Services: | Forum | Review | PDF | Favorites | |
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