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On the distribution of the deficit at ruin when claims are phase-type. (English) Zbl 1142.62088
The model under consideration is a general renewal risk model, for which insurer’s surplus is given by \(U(t)=u+ct-\sum_{i=1}^{N(T)} X_i\), where \(u\) is the initial capital, \(c\) is the rate of premium income, \(N(t)\) is a renewal claim number process, and the individual claim amounts \(\{X_i,i\geq 1\}\) are positive i.i.d. random variables. The purpose of this paper is to establish a fairy robust result regarding the distribution of the deficit at ruin in such a model given that ruin occurs.
The authors prove that if the claim amounts have a phase-type distribution, then so distributed they have a deficit, with precisely the same matrix of transition rates. The application of this result is illustrated with several examples. The necessary computations are carried out using standard software packages such as Mathematica and Maple.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
62E10 Characterization and structure theory of statistical distributions
65C60 Computational problems in statistics (MSC2010)
Software:
Maple; Mathematica
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[1] DOI: 10.1016/0167-6687(92)90058-J · Zbl 0748.62058 · doi:10.1016/0167-6687(92)90058-J
[2] DOI: 10.1214/aop/1176989805 · Zbl 0755.60049 · doi:10.1214/aop/1176989805
[3] DOI: 10.1142/9789812779311 · doi:10.1142/9789812779311
[4] Bowers NL, Actuarial Mathematics (1997)
[5] DOI: 10.1016/0167-6687(89)90028-0 · Zbl 0682.62083 · doi:10.1016/0167-6687(89)90028-0
[6] DOI: 10.1016/0167-6687(88)90100-X · Zbl 0637.62101 · doi:10.1016/0167-6687(88)90100-X
[7] DOI: 10.2143/AST.17.2.2014970 · doi:10.2143/AST.17.2.2014970
[8] Horváth A, Advances in algorithmic methods for stochastic models: Proceedings of the 3rd International Conference on Matrix Analytic Methods pp pp. 191–213– (2000)
[9] Kleinrock L, Queueing systems, Volume I: Theory, John Wiley & Sons (1975)
[10] DOI: 10.1137/1.9780898719734 · Zbl 0922.60001 · doi:10.1137/1.9780898719734
[11] DOI: 10.1016/S0167-6687(00)00038-X · Zbl 0971.91031 · doi:10.1016/S0167-6687(00)00038-X
[12] Neuts MF, Matrix-geometric solutions in stochastic models: An algorithmic approach, Johns Hopkins University Press (1981)
[13] DOI: 10.1002/9780470317044 · doi:10.1002/9780470317044
[14] DOI: 10.2143/AST.24.2.2005068 · doi:10.2143/AST.24.2.2005068
[15] DOI: 10.1080/034612300750066737 · Zbl 0958.91029 · doi:10.1080/034612300750066737
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