Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T09:45:56.375Z Has data issue: false hasContentIssue false
  • English
  • Français

Does Markov-Modulation Increase the Risk?

Published online by Cambridge University Press:  29 August 2014

Søren Asmussen ,
Andreas Frey ,
Tomasz Rolski  and
Volker Schmidt
Søren Asmussen*
Affiliation:
Aalborg University, Denmark
Andreas Frey*
Affiliation:
University of Ulm, Germany
Tomasz Rolski*
Affiliation:
University of Wroclaw, Poland
Volker Schmidt*
Affiliation:
University of Ulm, Germany
*
Institute of Electronic Systems, Aalborg University, Fr. Bajersv. 7, DK-9220 Aalborg, Denmark
Department of Stochastics, University of Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany
Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland
Department of Stochastics, University of Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we compare ruin functions for two risk processes with respect to stochastic ordering, stop-loss ordering and ordering of adjustment coefficients. The risk processes are as follows: in the Markov-modulated environment and the associated averaged compound Poisson model. In the latter case the arrival rate is obtained by averaging over time the arrival rate in the Markov modulated model and the distribution of the claim size is obtained by averaging the ones over consecutive claim sizes.

Keywords

Risk theorycompound Poisson modelMarkovian environmentstochastic orderingstop-loss orderingadjustment coefficientexponential tail orderingperturbation
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 25 , Issue 1 , May 1995 , pp. 49 - 66
Copyright
Copyright © International Actuarial Association 1995

References

[1]Asmussen, S. (1989) Risk theory in a Markovian environment. Scand. Actuarial J., 69–100.CrossRefGoogle Scholar
[2]Asmussen, S., Henriksen, L. Fløe and Klüppelberg, C. (1994) Large claims approximations for risk processes in a Markovian environment. Stock. Proc. Appt. 54, 29–43.CrossRefGoogle Scholar
[3]Asmussen, S. and Rolski, T. (1991). Computational methods in risk theory: A matrix-algorithmic approach, insurance: Mathematics and Economics 10, 259–274.Google Scholar
[4]Asmussen, S. and Rolski, T. (1994) Risk theory in a periodic environment; the Cramér-Lundberg approximation and Lundberg's inequality. Math. Oper. Res., 19, 410–433.CrossRefGoogle Scholar
[5]Asmussen, S. and Schmidt, V. (1995) Ladder height distributions with marks. Stoch. Proc. Appl. (to appear).Google Scholar
[6]Asmussen, S. (1994) Ruin Probabilities. Book manuscript, Aalborg University. To be published by World Science Publishers, Singapore.Google Scholar
[7]Chang, C., Chao, X.L. and Pinedo, M. (1991) Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture. Adv. Appl. Prob. 23, 210–228.CrossRefGoogle Scholar
[8]Chang, C. and Nelson, R. (1993) Perturbation analysis of the M/M/1 queue in a Markovian environment via the matrix-geometric method. Stochastic Models 9, 233–246.CrossRefGoogle Scholar
[9]Daley, D.J. and Rolski, T. (1984) Some comparability results for waiting times in single- and many-server queues. J. Appl. Prob. 21, 887–900.CrossRefGoogle Scholar
[10]Esary, J.D., Proschan, F. and Walkup, D.W. (1967) Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.CrossRefGoogle Scholar
[11]Franken, P., König, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. J. Wiley & Sons, Chichester.Google Scholar
[12]Goovaerts, M.J., Kaas, R., van Heerwarden, A.E. and Bauweunckx, T. ( 1990) Effective Actuarial Methods. North-Holland, Amsterdam.Google Scholar
[13]Grigelionis, B. (1993) Two-sided Lundberg inequalities in a Markovian environment. Liet. Matem. Rinkinys 33.Google Scholar
[14]Van Heerwarden, A.E. (1991) Ordering of Risks: Theory and Actuarial Applications. Tinbergen Institute Research Series 20, Amsterdam.Google Scholar
[15]Janssen, J. (1980) Some transient results on the M/SM/1 special semi-Markov model in risk and queueing theories. Astin Bulletin 11, 41–51.CrossRefGoogle Scholar
[16]Janssen, J. and Reinhard, J.M. (1985) Probabilities de ruine pour une classe de modèles de risque semi-Markoviens. Astin Bulletin 15, 123–133.CrossRefGoogle Scholar
[17]Kelly, F.P. (1979) Reversibility and Stochastic Networks. J. Wiley & Sons, Chichester.Google Scholar
[18]Klüppelberg, C. (1993) Asymptotic ordering of risks and ruin probabilities. Insurance: Mathematics and Economics 12, 259–264.Google Scholar
[19]König, D. and Schmidt, V. (1992) Random Point Processes. Teubner, Stuttgart (in Germ.).Google Scholar
[20]Makowski, A.M. (1994) On an elementary characterization of the increasing convex ordering, with an application. J. Appl. Prob. 31, 834–840.CrossRefGoogle Scholar
[21]Mitrinović, D.S., Pečarić, J.E. and Fink, A.M. (1993) Classical and Neuv Inequalities in Analysis. Kluwer Academic Publishers, Amsterdam.Google Scholar
[22]Neuts, M.F. (1978) The M/M/1 queue with randomly varying arrival rate and service rates. Opsearch, 155, 139–157Google Scholar
[23]Neuts, M.F. (1981) Matrix-Geometric Solutions in Stochastic Models. The John Hopkins University Press. Baltimore.Google Scholar
[24]Reinhard, J.M. (1984). On a class of semi-Markov risk models obtained as classical risk models in a markovian environment. Astin Bulletin 14, 23–43.CrossRefGoogle Scholar
[25]Rolski, T. (1981) Queues with non-stationary input stream: Ross's conjecture. Adv. Appl. Prob. 13, 603–618.CrossRefGoogle Scholar
[26]Rolski, T. (1989) Queues with nonstationary inputs. Queueing Systems 15, 113–129.CrossRefGoogle Scholar
[27]Ross, S.M. (1978) Average delay in queues with non-stationary Poisson arrivals. J. Appl. Prob. 8, 814–827.Google Scholar
[28]Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. J. Wiley & Sons, Chichester.Google Scholar