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Expansions for Markov-modulated systems and approximations of ruin probability

Published online by Cambridge University Press:  14 July 2016

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.

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 + βa1 + ·· ·+ (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.

MSC classification

Type
Research Papers
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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