Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-05T02:40:04.970Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Environment

Published online by Cambridge University Press:  29 August 2014

Jean-Marie Reinhard
Jean-Marie Reinhard*
Affiliation:
Groupe AG, Brussels, Belgium
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.

We consider a risk model in which the claim inter-arrivals and amounts depend on a markovian environment process. Semi-Markov risk models are so introduced in a quite natural way. We derive some quantities of interest for the risk process and obtain a necessary and sufficient condition for the fairness of the risk (positive asymptotic non-ruin probabilities). These probabilities are explicitly calculated in a particular case (two possible states for the environment, exponential claim amounts distributions).

Keywords

Semi-Markov processesruin theory
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 14 , Issue 1 , April 1984 , pp. 23 - 43
Copyright
Copyright © International Actuarial Association 1984

References

Cinlar, E. (1967). Time dépendance of queues with semi-markovian services. J. Appl. Prob., 4, 356–364. Queues with semi-markovian arrivals. J. Appl. Prob., 4, 365-379.CrossRefGoogle Scholar
Feller, W. (1971). An introduction to probability theory and its applications, Vol. II. Wiley: New York.Google Scholar
Janssen, J. (1970). Sur une généralisation du concept de promenade aléatoire sur la droite réelle. Ann. Inst. H. Poincaré, B, VI, 249–269).Google Scholar
Janssen, J. (1980). Some transient results on the M/SM/1 special semi-Markov model in risk and queuing theories. Astin Bull., 11, 41–51.CrossRefGoogle Scholar
Janssen, J. and Reinhard, J. M. (1982). Some duality results for a class of multivariate semi-Markov processes. J. Appl. Prob., March 1982.CrossRefGoogle Scholar
Miller, H. D. (1962). A matrix factorization problem in the theory of random variables defined on a finite Markov chain. Proc. Camb. Phil. Soc., 58, 268–285. Absorption probabilities for sums of random variables defined on a finite Markov chain. Proc. Camb. Phil. Soc., 286-298.CrossRefGoogle Scholar
Neuts, M. F. (1966). The single server queue with Poisson input and semi-Markov service times. J. Appl. Prob., 3, 202–230.CrossRefGoogle Scholar
Neuts, M. F. and Chen, Shun-Zer (1972). The infinite-server queue with Poisson arrivals and semi-Markov service times. J. Op. Res., 20, 425–433.CrossRefGoogle Scholar
Newbould, M. (1973). A classification of a random walk defined on a finite Markov chain. Z. Wahrsch. verw. Geb., 26, 95–104.CrossRefGoogle Scholar
Pyke, R. (1961a). Markov renewal processes: definitions and preliminary properties. Ann. Math. Stat., 32, 1231–1242.CrossRefGoogle Scholar
Pyke, R. (1961b). Markov renewal processes with finitely many states. Ann. Math. Stat., 1243–1259.CrossRefGoogle Scholar
Purdue, P. (1974). The M/M/1 queue in a markovian environment. J. Op. Res., 22, 562–569.CrossRefGoogle Scholar
Reinhard, J. M. (1981). A semi-markovian game of economic survival. Scand. Act. J., 1981, 23–38.CrossRefGoogle Scholar
Reinhard, J. M. (1982). Identités du type Baxter-Spitzer pour une classe de promenades aléatoires semi-markoviennes. To appear in Ann. Inst. H. Poincaré.Google Scholar
Seal, H. L. (1974). The numerical calculation of U(w, t), the probability of non-ruin in interval (0, t). Scand. Act. J., 1974, 121–139.CrossRefGoogle Scholar