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An insensitivity property of ladder height distributions

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

Michael Frenz  and
Volker Schmidt
Michael Frenz*
Affiliation:
Bergakademie Freiberg
Volker Schmidt*
Affiliation:
University of Ulm
*
∗Postal address: Bergakademie Freiberg, Fachbereich Mathematik, Bernhard-von-Cotta-Str. 2, DO-9200 Freiberg, Germany.
∗∗Postal address: Department of Statistics, University of Ulm, Helmholtzstrasse 18, DW-7900 Ulm, Germany.
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Abstract

This paper considers the undershoot of a general continuous-time risk process with dependent increments under a certain initial level. The increments are given by the locations and amounts of claims which are described by a stationary marked point process. Under a certain balance condition, it is shown that the distribution of the undershoot depends only on the mark distribution and on the intensity of the underlying point process, but not on the form of its distribution. In this way an insensitivity property is extended which has been proved in Björk and Grandell [3] for the ruin probability, i.e. for the probability that after a finite time interval the initial level will be crossed from above.

Keywords

RISK THEORYPOINT PROCESSDEPENDENT INCREMENTSRUIN PROBABILITY

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 616 - 624
Copyright
Copyright © Applied Probability Trust 1992 

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References

[1] Asmussen, S. (1987) Applied Probability and Queues. Wiley, Chichester.Google Scholar
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[3] Björk, T. and Grandell, J. (1985) An insensitivity property of the ruin probability. Scand. Actuarial J., 148–156.Google Scholar
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[8] König, D. and Schmidt, V. (1992) Random Point Processes. Teubner, Stuttgart (in German).Google Scholar
[9] Nieuwenhuis, G. (1989) Equivalence of functional limit theorems for stationary point processes and their Palm distributions. Prob. Theory Rel. Fields 81, 593–608.Google Scholar