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Approximations for the Gerber-Shiu expected discounted penalty function in the compound Poisson risk model. (English) Zbl 1122.60076

Summary: In the classical risk model with initial capital \(u\), let \(\tau(u)\) be the time of ruin, \(X_+(u)\) be the risk reserve just before ruin, and \(Y_+(u)\) be the deficit at ruin. H. U. Gerber and E. S. W. Shiu [N. Am. Actuar. J. 2, No. 3, 101–112 (1998; Zbl 1081.91528)] defined the function \(m_\delta(u)= E[e^{-\delta\tau(u)} w(X_+(u),Y_+(u))\mathbf{1} (\tau(u)<\infty)]\), where \(\delta\geq 0\) can be interpreted as a force of interest and \(w(r,s)\) as a penalty function, meaning that \(m_\delta(u)\) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for \(m_\delta(u)\) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired \(m_\delta(u)\) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations.

MSC:

60K10 Applications of renewal theory (reliability, demand theory, etc.)
91B30 Risk theory, insurance (MSC2010)
60K05 Renewal theory

Citations:

Zbl 1081.91528
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