zbMATH — the first resource for mathematics

A note on the compound binomial model with randomized dividend strategy. (English) Zbl 1193.91062
Summary: This paper considers the compound binomial model with randomized decisions on paying dividends. By using two formulas obtained by J. Y. Tan and X. Q. Yang [Insur. Math. Econ. 39, No. 1, 1–18 (2006; Zbl 1147.91349)], two defective renewal equations for the Gerber-Shiu penalty function are derived and solved. The analytic solutions obtained are utilized to derive the probability of ultimate ruin, the deficit distribution at ruin and the distribution of the claim causing ruin. The asymptotic estimate satisfied by the penalty function is discussed in some detail.

91B30 Risk theory, insurance (MSC2010)
60K05 Renewal theory
Full Text: DOI
[1] Albrecher, H.; Hartinger, J.; Tichy, R., On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier, Scandinavian actuarial journal, 2, 103-126, (2005) · Zbl 1092.91036
[2] Cheng, S.; Gerber, H.U.; Shiu, E.S.W., Discounted probabilities and ruin theory in the compound binomial model, Insurance: mathematics and economics, 26, 239-250, (2000) · Zbl 1013.91063
[3] B. De Finetti, Su un’impostazione alternativa della teoria collettiva del rischio, in: Proceedings of the Transactions of the XV International Congress of Actuaries, vol. 2, 1957, pp. 433-443.
[4] Dickson, D.C.M., Some comments on the compound binomial model, ASTIN bulletin, 24, 33-45, (1994)
[5] Dickson, D.C.M.; Waters, H., Some optimal dividend problems, ASTIN bulletin, 34, 49-74, (2004)
[6] Gerber, H.U., Mathematical fun with the compound binomial process, ASTIN bulletin, 18, 161-168, (1988)
[7] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 48-78, (1998) · Zbl 1081.60550
[8] H.U. Gerber, E.S.W. Shiu, On optimal dividend strategies in the compound Poisson model 2005. Preprint.
[9] Karlin, S.; Taylor, H.M., A first course in stochastic processes, (1975), Academic Press New York · Zbl 0315.60016
[10] Lin, X.; Pavlova, K.P., The compound Poisson risk model with a threshold dividend strategy, Insurance: mathematics and economics, 38, 57-80, (2006) · Zbl 1157.91383
[11] Lin, X.; Willmot, G.E., Analysis of a defective renewal equation arising in ruin theory, Insurance: mathematics and economics, 25, 63-84, (1999) · Zbl 1028.91556
[12] Lin, X.; Willmot, G.E.; Drekic, S., The classical risk model with a constant dividend barrier: analysis of the gerber – shiu discounted penalty function, Insurance: mathematics and economics, 33, 551-566, (2003) · Zbl 1103.91369
[13] Pavlova, K.P.; Willmot, G.E., The discrete stationary renewal risk model and the gerber – shiu discounted penalty function, Insurance: mathematics and economics, 35, 267-277, (2004) · Zbl 1103.91046
[14] Shiu, E.S.W., Probability of eventual ruin in the compound binomial model, ASTIN bulletin, 19, 179-190, (1989)
[15] Tan, J.Y.; Yang, X.Q., The compound binomial model with randomized decisions on paying dividends, Insurance: mathematics and economics, 39, 1-18, (2006) · Zbl 1147.91349
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.