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Erlangian Approximations for Finite-Horizon Ruin Probabilities

Published online by Cambridge University Press:  29 August 2014

Soren Asmussen ,
Florin Avram  and
Miguel Usabel
Soren Asmussen
Affiliation:
Mathematical Statistics, Centre for Mathematical Sciences, Box 118, 221 00 Lund, Sweden, E-mail: [email protected]
Florin Avram
Affiliation:
Dept. de Math., Universite de Pau and Dept. of Actuarial Maths. & Statistics, Heriot-Watt University, Edinburgh EH14 4AS, U.K., E-mail: [email protected]
Miguel Usabel
Affiliation:
Edif. Miguel de Unamuno, Universidad Carlos III, Avda. Universidad Carlos III 22, 28270 Colmenarejo (Madrid), Spain, E-mail: [email protected]
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Abstract

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For the Cramér-Lundberg risk model with phase-type claims, it is shown that the probability of ruin before an independent phase-type time H coincides with the ruin probability in a certain Markovian fluid model and therefore has an matrix-exponential form. When H is exponential, this yields in particular a probabilistic interpretation of a recent result of Avram & Usabel. When H is Erlang, the matrix algebra takes a simple recursive form, and fixing the mean of H at T and letting the number of stages go to infinity yields a quick approximation procedure for the probability of ruin before time T. Numerical examples are given, including a combination with extrapolation.

Keywords

Erlang expirationextrapolationfinite time ruin probabilitiesfluid modelphase-type distributionssemi-Markov embedding
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 32 , Issue 2 , November 2002 , pp. 267 - 281
Copyright
Copyright © International Actuarial Association 2002

References

1.Aldous, D. and Shepp, L. (1987) The least variable phase-type distribution is Erlang. Stochastic Models 3, 467–473.Google Scholar
2.Asmussen, S. (1989) Risk theory in a Markovian environment. Scand. Act. J. 89, 69–100.CrossRefGoogle Scholar
3.Asmussen, S. (1992) Phase-type representations in random walk and queueing problems. Ann. Probab. 20, 772–789.CrossRefGoogle Scholar
4.Asmussen, S. (1995a) Stationary distributions for fluid flow models with or without Brownian noise. Stochastic Models 11, 21–49.CrossRefGoogle Scholar
5.Asmussen, S. (1995b) Stationary distributions via first passage times. Advances in Queueing (Dshalalow, J., ed.), 79–102. CRC Press.Google Scholar
6.Asmussen, S. (2000) Matrix-analytic models and their analysis. Scand. J. Statist. 27, 193–226.CrossRefGoogle Scholar
7.Asmussen, S. (2000) Ruin Probabilities. World Scientific.CrossRefGoogle Scholar
8.Asmussen, S. and Bladt, M. (1996) Renewal theory and queueing algorithms for matrix-exponential distributions. Matrix-Analytic Methods in Stochastic Models (Alfa, A.S. & Chakravarty, S., eds.), 313–341. Marcel Dekker, New York.Google Scholar
9.Asmussen, S. and Kella, O. (2000) A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Probab. 32, 376–393.CrossRefGoogle Scholar
10.Asmussen, S. and Rolski, T. (1991) Computational methods in risk theory: a matrix-algorithmic approach. Insurance: Mathematics and Economics 10, 259–274.Google Scholar
11.Avram, F. and Usabel, M. (2001) Finite time ruin probabilities with one Laplace inversion. Submitted to Insurance: Mathematics and Economics.Google Scholar
12.Avram, F. and Usabel, M. (2001) Ruin probabilities and deficit for the renewal risk model with phase-type interarrival times. Submitted to Scand. Act. J. 19.Google Scholar
13.Avram, F. and Usabel, M. (2002) An ordinary differential equations approach for finite time ruin probabilities, including interest rates. Submitted to Insurance: Mathematics and Economics.Google Scholar
14.Barlow, M.T., Rogers, L.C.G. and Williams, D. (1980) Wiener-Hopf factorization for matrices, in Seminaire de Probabilites XIV, Lecture Notes in Math. 784, Springer, Berlin, 324–331.Google Scholar
15.Gerber, H., Goovaerts, M. and Kaas, R. (1987) On the probability and severity of ruin. Astin Bulletin 17, 151–163.CrossRefGoogle Scholar
16.Graham, A. (1981) Kronecker Products and Matrix Calculus. Wiley.Google Scholar
17.Latouche, G. and Ramaswami, V. (1999) Introduction to Matrix-Analytic Methods in Stochstic Modelling. SIAM.Google Scholar
18.London, R.R., Mckean, H.P., Rogers, L.C.G. and Williams, D. (1982) A martingale approach to some Wiener-Hopf problems II, in Seminaire de Probabilites XVI, Lecture Notes in Math. 920, Springer, Berlin, 68–90.Google Scholar
19.Mitra, D. (1988), Stochastic Fluid Models, Performance 87, Courtois, P. J. and Latouche, G. (editors), Elsevier, (North-Holland), 39–51.Google Scholar
20.Neuts, M.F. (1981) Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore. London.Google Scholar
21.Neuts, M.F. (1989) Structured Stochastic Matrices of the MIGI1 Type and Their Applications. Marcel Dekker.Google Scholar
22.Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. (1986) Numerical recipes. Cambridge University Press.Google Scholar
23.Standford, D.A. and Stroinski, K.J. (1994) Recursive method for computing finite-time ruin probabilities for phase-distributed claim sizes. Astin Bulletin 24, 235–254.CrossRefGoogle Scholar
24.Usabel, M. (1999) Calculating multivariate ruin probabilities via Gaver-Stehfest inversion technique. Insurance: Mathematics and Economics 25, 133–142.Google Scholar
25.Wikstad, N. (1971) Exemplification of ruin probabilities. Astin Bulletin 7, 147–152.CrossRefGoogle Scholar