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On the infinite-horizon probability of (non)ruin for integer-valued claims

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Zvetan G. Ignatov  and
Vladimir K. Kaishev
Zvetan G. Ignatov*
Affiliation:
Sofia University ‘St Kliment Ohridski’
Vladimir K. Kaishev*
Affiliation:
City University, London
*
∗Postal address: Faculty of Economics and Business Administration, Sofia University ‘St Kliment Ohridski’, 125 Tsarigradsko Shosse Boulevard, bl. 3, Sofia 1113, Bulgaria.
∗∗Postal address: Faculty of Actuarial Science and Statistics, Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, UK. Email address: [email protected]
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Abstract

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We consider a compound Poisson process whose jumps are modelled as a sequence of positive, integer-valued, dependent random variables, W1,W2,…, viewed as insurance claim amounts. The number of points up to time t of the stationary Poisson process which models the claim arrivals is assumed to be independent of W1,W2,…. The premium income to the insurance company is represented by a nondecreasing, nonnegative, real-valued function h(t) on [0,∞) such that limt→∞h(t) = ∞. The function h(t) is interpreted as an upper boundary. The probability that the trajectory of such a compound Poisson process will not cross the upper boundary in infinite time is known as the infinite-horizon nonruin probability. Our main result in this paper is an explicit expression for the probability of infinite-horizon nonruin, assuming that certain conditions on the premium-income function, h(t), and the joint distribution of the claim amount random variables, W1,W2,…, hold. We have also considered the classical ruin probability model, in which W1,W2,… are assumed to be independent, identically distributed random variables and we let h(t)=u + ct. For this model we give a formula for the nonruin probability which is a special case of our main result. This formula is shown to coincide with the infinite-horizon nonruin probability formulae of Picard and Lefèvre (2001), Gerber (1988), (1989), and Shiu (1987), (1989).

Keywords

Infinite-time (non)ruin probabilityfirst crossing of an upper boundary problemrisk processPoisson claim arrivaldependent claim amounts

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60K99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 535 - 551
Copyright
© Applied Probability Trust 2006 

References

Asmussen, S. (2000). Ruin Probabilities. World Scientific, River Edge, NJ.Google Scholar
Fichtenholz, G. M. (1969). Differential and Integral Calculus, Vol. 2. Fizmatgiz, Moscow (in Russian).Google Scholar
Gerber, H. U. (1979). An Introduction to Mathematical Risk Theory. Huebner, Philadelphia, PA.Google Scholar
Gerber, H. U. (1988). Mathematical fun with ruin theory. Insurance Math. Econom. 7, 15–23.CrossRefGoogle Scholar
Gerber, H. U. (1989). From the convolution of uniform distributions to the probability of ruin. Schweiz. Verein. Versicherungsmath. Mitt. 1989, 283–292.Google Scholar
Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.Google Scholar
Ignatov, Z. G. and Kaishev, V. K. (2000). Two-sided bounds for the finite time probability of ruin. Scand. Actuarial J. 2000, 46–62.CrossRefGoogle Scholar
Ignatov, Z. G., Kaishev, V. K. and Krachunov, R. S. (2001). An improved finite-time ruin probability formula and its Mathematica implementation. Insurance Math. Econom. 29, 375–386.Google Scholar
Picard, P. and Lefèvre, C. (2001). On the probability of (non-) ruin in infinite time. Scand. Actuarial J. 2001, 148–161.Google Scholar
Shiu, E. S. W. (1987). Convolution of uniform distributions and ruin probability. Scand. Actuarial J. 1987, 191–197.Google Scholar
Shiu, E. S. W. (1989). Ruin probability by operational calculus. Insurance Math. Econom. 8, 243–249.CrossRefGoogle Scholar