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Erlangian Approximations for Finite-Horizon Ruin Probabilities

  • Soren Asmussen (a1), Florin Avram (a2) and Miguel Usabel (a3)

Abstract

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.

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References

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1.Aldous, D. and Shepp, L. (1987) The least variable phase-type distribution is Erlang. Stochastic Models 3, 467473.
2.Asmussen, S. (1989) Risk theory in a Markovian environment. Scand. Act. J. 89, 69100.
3.Asmussen, S. (1992) Phase-type representations in random walk and queueing problems. Ann. Probab. 20, 772789.
4.Asmussen, S. (1995a) Stationary distributions for fluid flow models with or without Brownian noise. Stochastic Models 11, 2149.
5.Asmussen, S. (1995b) Stationary distributions via first passage times. Advances in Queueing (Dshalalow, J., ed.), 79102. CRC Press.
6.Asmussen, S. (2000) Matrix-analytic models and their analysis. Scand. J. Statist. 27, 193226.
7.Asmussen, S. (2000) Ruin Probabilities. World Scientific.
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.), 313341. Marcel Dekker, New York.
9.Asmussen, S. and Kella, O. (2000) A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Probab. 32, 376393.
10.Asmussen, S. and Rolski, T. (1991) Computational methods in risk theory: a matrix-algorithmic approach. Insurance: Mathematics and Economics 10, 259274.
11.Avram, F. and Usabel, M. (2001) Finite time ruin probabilities with one Laplace inversion. Submitted to Insurance: Mathematics and Economics.
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.
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.
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, 324331.
15.Gerber, H., Goovaerts, M. and Kaas, R. (1987) On the probability and severity of ruin. Astin Bulletin 17, 151163.
16.Graham, A. (1981) Kronecker Products and Matrix Calculus. Wiley.
17.Latouche, G. and Ramaswami, V. (1999) Introduction to Matrix-Analytic Methods in Stochstic Modelling. SIAM.
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, 6890.
19.Mitra, D. (1988), Stochastic Fluid Models, Performance 87, Courtois, P. J. and Latouche, G. (editors), Elsevier, (North-Holland), 3951.
20.Neuts, M.F. (1981) Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore. London.
21.Neuts, M.F. (1989) Structured Stochastic Matrices of the MIGI1 Type and Their Applications. Marcel Dekker.
22.Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. (1986) Numerical recipes. Cambridge University Press.
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, 235254.
24.Usabel, M. (1999) Calculating multivariate ruin probabilities via Gaver-Stehfest inversion technique. Insurance: Mathematics and Economics 25, 133142.
25.Wikstad, N. (1971) Exemplification of ruin probabilities. Astin Bulletin 7, 147152.

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