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Erlangian approximations for finite-horizon ruin probabilities. (English) Zbl 1081.60028

The authors consider the probability \(\psi(u,T)\) of ruin before time \(T>0\) in the Cramér-Lundberg risk model, where \(u\) denotes the initial capital. F. Avram and M. Usabel [Insur. Math. Econ. 32, No. 3, 371–377 (2003; Zbl 1074.91026)] have shown that, if the individual claim amount distribution is of phase-type, and if \(T\) is an independent random variable with an exponential distribution, then \(\psi(u,T)\) can be evaluated via a matrix-exponential formula. In the present paper, an extension is given, when the distribution of \(T\) is of phase-type. It is shown, how the case of Erlang distributed \(T\) can be used to approximate \(\psi(u,T_0)\) for fixed \(T_0\).

MSC:

60G51 Processes with independent increments; Lévy processes
60K15 Markov renewal processes, semi-Markov processes
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
91B30 Risk theory, insurance (MSC2010)

Citations:

Zbl 1074.91026
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References:

[1] Structured Stochastic Matrices of the MIGI1 Type and Their Applications (1989)
[2] Numerical recipes (1986) · Zbl 0587.65005
[3] DOI: 10.1239/aap/1013540169 · Zbl 0961.60081 · doi:10.1239/aap/1013540169
[4] Matrix-Analytic Methods in Stochastic Models pp 313– (1996)
[5] Ruin Probabilities (2000) · Zbl 0986.62086
[6] DOI: 10.1111/1467-9469.00186 · Zbl 0959.60085 · doi:10.1111/1467-9469.00186
[7] Advances in Queueing pp 79– (1995)
[8] DOI: 10.1080/15326349508807330 · Zbl 0817.60086 · doi:10.1080/15326349508807330
[9] DOI: 10.1214/aop/1176989805 · Zbl 0755.60049 · doi:10.1214/aop/1176989805
[10] Scand. Act. J. 89 pp 69– (1989)
[11] DOI: 10.1080/15326348708807067 · Zbl 0635.60086 · doi:10.1080/15326348708807067
[12] Stochastic Fluid Models, Performance 87 pp 39– (1988)
[13] A martingale approach to some Wiener-Hopf problems II, in Seminaire de Probabilites XVI pp 68– (1982) · Zbl 0485.60073
[14] Introduction to Matrix-Analytic Methods in Stochstic Modelling (1999)
[15] Kronecker Products and Matrix Calculus (1981) · Zbl 0497.26005
[16] DOI: 10.2143/AST.17.2.2014970 · doi:10.2143/AST.17.2.2014970
[17] Wiener-Hopf factorization for matrices, in Seminaire de Probabilites XIV 784 pp 324– (1980) · Zbl 0429.15007
[18] Scand. Act. J. 19 (2001)
[19] Insurance: Mathematics and Economics 10 pp 259– (1991)
[20] Astin Bulletin 7 pp 147– (1971)
[21] Insurance: Mathematics and Economics 25 pp 133– (1999)
[22] DOI: 10.2143/AST.24.2.2005068 · doi:10.2143/AST.24.2.2005068
[23] Matrix-Geometric Solutions in Stochastic Models (1981)
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