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Expansions for Markov-modulated systems and approximations of ruin probability

Part of: Stochastic processes Special 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) = a 0 + β ∗ a 1 + ·· ·+ (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 a 1 and a 2 for the probability of ruin φ u with initial capital u for the risk process in the Markov-modulated environment; a 0 = 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

Information

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.

References

Asmussen, S. (1989) Risk theory in a Markovian environment. Scand. Actuarial J, 69–100.CrossRefGoogle Scholar
Asmussen, S. (1991a) Ladder heights and the Markov-modulated M/G/1 queues. Stoch. Proc. Appl. 37, 313–326.CrossRefGoogle Scholar
Asmussen, S. (1991b) Light traffic equivalence in single-server queues. Ann. Appl. Prob. 2, 555–574.Google Scholar
Asmussen, S. (1993) Fundamentals of Ruin Probability Theory. Book manuscript.Google Scholar
Asmussen, S. and Bladt, M. (1993) Phase-type distributions and risk processes with state-dependent premiums. Preprint, University of Aalborg. Google Scholar
Asmussen, S, and Schock Petersen, S. (1988) Ruin probabilities expressed in terms of storage processes. Adv. Appl. Prob. 20, 913–916.CrossRefGoogle Scholar
Baccelli, F. and Brémaud, P. (1993) Virtual customers in sensitivity and light traffic analysis via Campbell's formula for point processes. Adv. Appl. Prob. 25, 221–234.CrossRefGoogle Scholar
Blaszczyszyn, B. (1995) Factorial moment expansion for stochastic systems. Stoch. Proc. Appl. 56, 321–335.CrossRefGoogle Scholar
Blaszczyszyn, B. and Rolski, T. (1993) Queues in series in light traffic. Ann. Appl. Prob. 3, 881–896.CrossRefGoogle Scholar
Brémaud, P. (1992) Private communication.Google Scholar
Burman, D. Y. and Smith, D. R. (1986) An asymptotic analysis of queueing systems with Markov modulated arrivals. Operat. Res. 34, 105–119.CrossRefGoogle Scholar
Daley, D. J. and Rolski, T. (1991) Light traffic approximations in queues. Math. Operat. Res. 16, 57–71.CrossRefGoogle Scholar
Daley, D. J. and Rolski, T. (1992) Light traffic approximations in many-server queues. Adv. Appl. Prob. 24, 202–218.CrossRefGoogle Scholar
Daley, D. J. and Rolski, T. (1994) Light traffic approximations in general stationary single-server queues. Stoch. Proc. Appl. 49, 141–158.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, (1988) An Introduction to the Theory of Point Processes, Springer, New York.Google Scholar
Grandell, J. (1976) Doubly Stochastic Poisson Processes. (Lecture Notes in Mathematics 529). Springer, Berlin.CrossRefGoogle Scholar
Harrrison, N. U. and Resnick, S. I. (1978) The recurrence classification of risk and storage processes. Math. Operat. Res. 3, 57–66.CrossRefGoogle Scholar
Kallenberg, O. (1983) Random Measures. 3rd edn. Academic Press, New York.CrossRefGoogle Scholar
Karr, A. F. (1986) Point Processes and their Statistical Inference. Dekker, New York.Google Scholar
Kroese, D. P. and Schmidt, V. (1995) Light-traffic analysis for queues with spatially distributed arrivals. Math. Operat. Res. 20.Google Scholar
Neuts, M. (1978) The M/M/1 queue with randomly varying rate and service rates. Opsearch 155, 139–157.Google Scholar
Reiman, M. I. and Simon, B. (1989) Open queueing systems in light traffic. Math. Operat. Res. 14, 26–59.CrossRefGoogle Scholar
Reiman, M. I. and Weiss, A. (1989) Light traffic derivatives via likelihood ratios. IEEE Trans. Inform. Theory 35, 648–654.CrossRefGoogle Scholar
Rolski, T. (1992) Approximations of performance characteristics in periodic Poisson queues. In Queueing and Related Models, pp. 285–298. ed. Bhat, U. N. and Basava, I. V. Clarendon Press, Oxford.Google Scholar
Rudemo, M. (1973) Point processes generated by transitions of Markov chains. Adv. Appl. Prob. 5, 262–282.CrossRefGoogle Scholar
Ryll-Nardzewski, C. (1961) Remarks on processes of calls. Math. Statist. Prob. 2, 455–465.Google Scholar
Sigman, K. (1992) Light traffic for work-load in queues. Queueing Systems 11, 429–442.CrossRefGoogle Scholar
Zazanis, M. A. (1992) Analyticity of Poisson-driven stochastic systems. Adv. Appl. Prob. 24, 532–541.CrossRefGoogle Scholar