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Expansions for Markov-modulated systems and approximations of ruin probability

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Bartłomiej Błaszczyszyn  and
Tomasz Rolski
Bartłomiej Błaszczyszyn*
Affiliation:
University of Wroclaw
Tomasz Rolski*
Affiliation:
University of Wroclaw
*
∗Postal address for both authors: Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50–384 Wroclaw, Poland.
∗Postal address for both authors: Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50–384 Wroclaw, Poland.
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Abstract

Let N be a stationary Markov-modulated marked point process on ℝ with intensity β∗ and consider a real-valued functional ψ(N). In this paper we study expansions of the form Eψ(N) = a0 + β∗a1 + ·· ·+ (β∗)nan + o((β∗)n) for β∗→ 0. Formulas for the coefficients ai are derived in terms of factorial moment measures of N. We compute a1 and a2 for the probability of ruin φ u with initial capital u for the risk process in the Markov-modulated environment; a0 = 0. Moreover, we give a sufficient condition for ϕu to be an analytic function of β∗. We allow the premium rate function p(x) to depend on the actual risk reserve.

Keywords

RUIN PROBABILITYn-FOLD PALM DISTRIBUTIONSCAMPBELL MEASUREFACTORIAL MOMENT MEASUREMARKOV-MODULATED MARKED POINT PROCESSGENERAL PREMIUM RATEEXPANSIONANALYTICITY

MSC classification

Primary: 60G55: Point processes
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 57 - 70
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

Work supported by KBN under grant 2 1023 90 01.

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