Discrete Lundberg-type bounds with actuarial applications
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 217-235.

Different kinds of renewal equations repeatedly arise in connection with renewal risk models and variations. It is often appropriate to utilize bounds instead of the general solution to the renewal equation due to the inherent complexity. For this reason, as a first approach to construction of bounds we employ a general Lundberg-type methodology. Second, we focus specifically on exponential bounds which have the advantageous feature of being closely connected to the asymptotic behavior (for large values of the argument) of the renewal function. Finally, the last section of this paper includes several applications to risk theory quantities.

DOI : 10.1051/ps:2007016
Classification : 62E99, 60G51, 62P05
Mots-clés : deficit at ruin, discrete renewal equation, probability of ultimate ruin, stop-loss premium, surplus immediately before ruin 
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     author = {Sendova, Kristina},
     title = {Discrete {Lundberg-type} bounds with actuarial applications},
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     mrnumber = {2320817},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2007016/}
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Sendova, Kristina. Discrete Lundberg-type bounds with actuarial applications. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 217-235. doi : 10.1051/ps:2007016. http://www.numdam.org/articles/10.1051/ps:2007016/

[1] H. Bühlmann, Mathematical Methods in Risk Theory. Springer, New York (1970). | MR | Zbl

[2] E. Cossette and E. Marceau, The discrete-time risk model with correlated classes of business. Insurance: Mathematics and Economics 26 (2000) 133-149. | Zbl

[3] S.N. Ethier and D. Khoshnevisan, Bounds on gambler's ruin probabilities in terms of moments. Methodology and Computing in Applied Probability 4 (2002) 55-68. | Zbl

[4] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edition John Wiley, New York (1968). | MR | Zbl

[5] H.U. Gerber and E.S.W. Shiu, On the time value of ruin. North American Actuarial Journal 1 (1998) 48-78. | Zbl

[6] S. Li, On a class of discrete time renewal risk models. Scandinavian Actuarial Journal 4 (2005a) 241-260.

[7] S. Li, Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time risk models. Scandinavian Actuarial Journal 4 (2005b) 271-284.

[8] K.P. Pavlova, Some Aspects of Discrete Ruin Theory. Ph.D. Thesis. University of Waterloo, Canada (2004).

[9] K.P. Pavlova and G.E. Willmot, The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function. Insurance: Mathematics and Economics 35 (2004) 267-277. | Zbl

[10] K.P. Pavlova, G.E. Willmot and J. Cai, Preservation under convolution and mixing of some discrete reliability classes. Insurance: Mathematics and Economics 38 (2006) 391-405. | Zbl

[11] P. Picard and C. Lefèvre, The moments of ruin time in the classical risk model with discrete claim size distribution. Insurance: Mathematics and Economics 23 (1998) 157-172. | Zbl

[12] P. Picard and C. Lefèvre, The probability of ruin in finite time with discrete claim size distribution. Scandinavian Actuarial Journal 1 (1997) 58-69. | Zbl

[13] S. Resnick, Adventures in Stochastic Processes. Birkäuser, Boston (1992). | MR | Zbl

[14] S. Ross, Stochastic Processes. 2nd edition, John Wiley, New York (1996). | MR | Zbl

[15] M. Shaked and J. Shanthikumar, Stochastic Orders and their Applications. Academic Press, San Diego, CA (1994). | MR | Zbl

[16] G.E. Willmot, On higher-order properties of compound geometric distributions, J. Appl. Probab. 39 (2002a) 324-340. | Zbl

[17] G.E. Willmot, Compound geometric residual lifetime distributions and the deficit at ruin. Insurance: Mathematics and Economics 30 (2002b) 421-438. | Zbl

[18] G.E. Willmot, J. Cai, Aging and other distributional properties of discrete compound geometric distributions. Insurance: Mathematics and Economics 28 (2001) 361-379. | Zbl

[19] G.E. Willmot, J. Cai and X.S. Lin, Lundberg inequalities for renewal equations. Adv. Appl. Probab. 33 (2001) 674-689. | Zbl

[20] G.E. Willmot, S. Drekic and J. Cai, Equilibrium compound distributions and stop-loss moments. Scandinavian Actuarial Journal 1 (2005) 6-24.

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