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Lundberg inequalities for a Cox model with a piecewise constant intensity

Part of: Stochastic processes Stochastic analysis Markov processes

Published online by Cambridge University Press:  14 July 2016

Hanspeter Schmidli
Hanspeter Schmidli*
Affiliation:
Heriot-Watt University
*
∗Postal address: Actuarial Maths & Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK.
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Abstract

A Cox risk process with a piecewise constant intensity is considered where the sequence (Li) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

Keywords

COX MODELRUIN PROBABILITYLUNDBERG INEQUALITYCRAMER–LUNDBERG APPROXIMATIONRISK THEORYMARTINGALE METHODSCHANGE OF MEASURE

MSC classification

Primary: 60H99: None of the above, but in this section
Secondary: 60G44: Martingales with continuous parameter 60J99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 196 - 210
Copyright
Copyright © Applied Probability Trust 1996 

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References

Ammeter, H. (1948) A generalization of the collective theory of risk in regard to fluctuating basic probabilities. Skand. Akt. Tidskr., 171–198.Google Scholar
Asmussen, S. (1989) Risk theory in a Markovian environment. Scand. Act. J, 66–100.CrossRefGoogle Scholar
Asmussen, S. (1994) Ruin Probabilities. Book manuscript, Aalborg University.Google Scholar
Björk, T. and Grandell, J. (1988) Exponential inequalities for ruin probabilities in the Cox case. Scand. Act. J., 77–111.CrossRefGoogle Scholar
Dassios, A. and Embrechts, P. (1980) Martingales and insurance risk. Commun. Statist. Stoch. Models 5, 181–217.Google Scholar
Davis, M. H. A. (1984) Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.Google Scholar
Davis, M. H. A. (1993) Markov Models and Optimization. Chapman and Hall, London.Google Scholar
Dunford, N. and Schwartz, J. T. (1958) Linear Operators, Part I. Interscience, New York.Google Scholar
Embrechts, P., Grandell, J. and Schmidli, H. (1993) Finite-time Lundberg inequalities in the Cox case. Scand. Act. J., 17–41.CrossRefGoogle Scholar
Gerber, H. U. (1973) Martingales in risk theory. Schweiz. Verein. Versicherungsmath. Mitt. 73, 205–216.Google Scholar
Grandell, J. (1991) Aspects of Risk Theory. Springer, New York.CrossRefGoogle Scholar
Hochstadt, H. (1973) Integral Equations. Wiley, New York.Google Scholar
Kingman, J. F. C. (1981) A convexity property of positive matrices. Quart. J. Math. Oxford Ser. (2) 12, 283–284.Google Scholar
Schaefer, H. H. (1974) Banach Lattices and Positive Operators. Springer, Berlin.CrossRefGoogle Scholar
Schmidli, H. (1994) Cramer-Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion. Research Report No. 291. Dept. Theor. Statist., Aarhus University.Google Scholar