×

zbMATH — the first resource for mathematics

Approximations for moments of deficit at ruin with exponential and subexponential claims. (English) Zbl 1092.62599
Summary: Consider a renewal insurance risk model with initial surplus \(u>0\) and let \(A_u\) denote the deficit at the time of ruin. This paper investigates the asymptotic behavior of the moments of \(A_u\) as \(u\) tends to infinity. Under the assumption that the claim size is exponentially or subexponentially distributed, we obtain some asymptotic relationships for the \(\phi\)-moments of \(A_u\), where \(\phi\) is a non-negative and non-decreasing function satisfying certain conditions.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
62E20 Asymptotic distribution theory in statistics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bingham, N.H.; Goldie, C.M.; Teugels, J.L., Regular variation, (1987), Cambridge University Press Cambridge
[2] Dickson, D.C.M.; Waters, H.R., The probability and severity of ruin in finite and in infinite time, ASTIN bull., 22, 177-190, (1992)
[3] Dufresne, F.; Gerber, H.U., The probability and severity of ruin for combinations of exponential claim amount distributions and their translations, Insurance: math. econom., 7, 75-80, (1988) · Zbl 0637.62101
[4] Dufresne, F.; Gerber, H.U., The surpluses immediately before and at ruin, and the amount of the claim causing ruin, Insurance: math. econom., 7, 193-199, (1988) · Zbl 0674.62072
[5] Embrechts, P.; Veraverbeke, N., Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance: math. econom., 1, 55-72, (1982) · Zbl 0518.62083
[6] Feller, W., 1971. An Introduction to Probability Theory and Its Applications, 2nd Edition, Vol. 2. Wiley, New York. · Zbl 0219.60003
[7] Gerber, H.U.; Goovaerts, M.J.; Kaas, R., On the probability and severity of ruin, Astin bull., 17, 2, 151-163, (1987)
[8] Gerber, H.U.; Shiu, E.S.W., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: math. econom., 21, 129-137, (1997) · Zbl 0894.90047
[9] Kalashnikov, V.V., Two-sided bounds for ruin probabilities, Scand. act. J., 1, 1-18, (1996) · Zbl 0845.62075
[10] Kl├╝ppelberg, C., Subexponential distributions and characterization of related classes, Probab. theory related fields, 82, 259-269, (1989) · Zbl 0687.60017
[11] Lin, X.S.; Willmot, G.E., The moments of the time of ruin, the surplus before ruin, and the deficit at ruin, Insurance: math. econom., 27, 1, 19-44, (2000) · Zbl 0971.91031
[12] Schmidli, H., On the distribution of the surplus prior to and at ruin, ASTIN bull., 29, 227-244, (1999) · Zbl 1129.62425
[13] Tang, Q.H., 2001. Extremal values of risk processes for insurance and finance: with special emphasis on the possibility of large claims. Doctoral Thesis, University of Science and Technology of China.
[14] Tang, Q.H., 2002. Moments of the deficit at ruin in the renewal model with heavy-tailed claims. Working paper, to appear.
[15] Veraverbeke, N., Asymptotic behavior of weiner – hopf factors of a random walk, Stochastic proc. appl., 5, 27-37, (1977) · Zbl 0353.60073
[16] Yang, H.; Zhang, L.H., The joint distribution of surplus immediately before ruin and the deficit at ruin under interest force, North American actuarial J., 5, 3, 92-103, (2001) · Zbl 1083.62547
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.