# zbMATH — the first resource for mathematics

Tail bounds for the distribution of the deficit in the renewal risk model. (English) Zbl 1189.91080
Summary: We obtain upper and lower bounds for the tail of the deficit at ruin in the renewal risk model, which are (i) applicable generally; and (ii) based on reliability classifications. We also derive two-side bounds, in the general case where a function satisfies a defective renewal equation, and we apply them to the renewal model, using the function $$\Lambda_u$$ introduced by G. Psarrakos and K. Politis [Stoch. Models 25, No. 1, 90–109 (2009; Zbl 1159.91412)]. Finally, we construct an upper bound for the integrated function $$\int_y^{\infty} \Lambda_u(z)\text dz$$ and an asymptotic result when the adjustment coefficient exists.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60K10 Applications of renewal theory (reliability, demand theory, etc.) 62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text:
##### References:
 [1] Apostol, T.M., Mathematical analysis, (1974), Addison-Wesley Hong Kong · Zbl 0126.28202 [2] Asmussen, S., Applied probability and queues, (1987), Wiley New York · Zbl 0624.60098 [3] Asmussen, S., Ruin probabilities, (2000), World Scientific Singapore [4] Bowers, N.; Gerber, H.; Hickman, J.; Jones, D.; Nesbitt, C., Actuarial mathematics, (1986), Society of Actuaries Ithaca · Zbl 0634.62107 [5] Brown, M., Error bounds for exponential approximation of geometric convolutions, Annals of probability, 18, 1388-1402, (1990) · Zbl 0709.60016 [6] Cai, J.; Garrido, J., Aging properties and bounds for ruin probabilities and stop-loss premiums, Insurance: mathematics and economics, 23, 33-43, (1998) · Zbl 0954.62123 [7] Chadjiconstantinidis, S.; Politis, K., Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model, Insurance: mathematics and economics, 41, 41-52, (2007) · Zbl 1119.91050 [8] Gerber, H.U.; Goovaerts, M.J.; Kaas, R., On the probability and severity of ruin, ASTIN bulletin, 17, 151-163, (1987) [9] Gerber, H.U.; Shiu, E.S.W., The time value of ruin in a sparre Andersen model, North American actuarial journal, 9, 2, 49-69, (2005), Discussions: 69-84 [10] De Vylder, F.; Goovaerts, M., Bounds for classical ruin probabilities, Insurance: mathematics and economics, 3, 121-131, (1984) · Zbl 0547.62068 [11] Feller, W., An introduction to probability theory and its applications, volume II, (1971), Addison-Wesley New York · Zbl 0219.60003 [12] Psarrakos, G., Politis, K., 2007. A generalisation of the Lundberg condition in the Sparre Andersen model and some applications (submitted for publication) · Zbl 1159.91412 [13] Szekli, R., On the concavity of the waiting time distribution in some GI/G/1 queues, Journal of applied probability, 23, 555-561, (1986) · Zbl 0599.60085 [14] Willmot, G.E., Compound geometric residual lifetime distributions and the deficit at ruin, Insurance: mathematics and economics, 30, 421-438, (2002) · Zbl 1039.62097 [15] Willmot, G.E.; Lin, X.S., Lundberg approximations for compound distributions with insurance applications, (2001), Springer New York
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.