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Homogeneous risk models with equalized claim amounts. (English) Zbl 1103.91361

Summary: We consider an homogeneous risk model on a fixed bounded time interval \([0, t]\) and we denote by \(N_t\) the number of claims in that interval. The claim amounts are \(X_1,X_2,\dots,X_{N_t}\). The homogeneous model is an extension of the classical actuarial risk model with \(N_t\) not necessarily Poisson distributed. In the model with equalized claim amounts, each amount \(X_k\) is replaced with \(X_k^\sim=(X_1+\cdots +X_{N_t})/N_t\). Let \(\Psi(t,u)\) be the ruin probability before \(t\) in the homogeneous model, corresponding to the initial risk reserve \(u\geq 0\) and let \(\Psi\not|\sim (t,u)\) be the corresponding ruin probability evaluated in the associated model with equalized claim amounts. The essence of the classical Prabhu formula is that \(\Psi(t,0)=\Psi^\sim (t,0)\). By rather systematic numerical investigations in the classical risk model, we verify that \(\Psi^\sim (t,u)\leq \Psi(t, u)\) for any value of \(u\geq 0\) and that \(\Psi^\sim (t, u)\) is an excellent approximation of \(\Psi(t,u)\). Then these conclusions must be valid in any homogeneous model and this is an interesting observation because \(\Psi^\sim(t,u)\) can be calculated numerically, whereas no algorithms are yet available for the numerical evaluation of \(\Psi(t,u)\) in general homogeneous risk models.

MSC:

91B30 Risk theory, insurance (MSC2010)
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References:

[1] De Vylder, F.E., 1996. Advanced Risk Theory. A Self-Contained Introduction. Editions de l’Université de Bruxelles, Swiss Association of Actuaries.; De Vylder, F.E., 1996. Advanced Risk Theory. A Self-Contained Introduction. Editions de l’Université de Bruxelles, Swiss Association of Actuaries.
[2] De Vylder, F. E.; Goovaerts, M., Inequality extensions of Prabhu’s formula in ruin theory, Insurance: Mathematics Economics, 24, 3, 249-272 (1999) · Zbl 0982.91032
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