×

zbMATH — the first resource for mathematics

A threshold-based risk process with a waiting period to pay dividends. (English) Zbl 1412.60064
Summary: In this paper, a modified dividend strategy is proposed by delaying dividend payments until the insurer’s surplus level remains at or above a threshold level \(b\) for a predetermined period of time \(h\). We consider two cases depending on whether the period of time sustained at or above level \(b\) is counted either consecutively or accumulatively (referred to as standard or cumulative waiting period). In both cases, we develop a recursive computational procedure to calculate the expected total discounted dividend payments made prior to ruin for a discrete-time Sparre Andersen renewal risk process. By varying the values of \(b\) and \(h\), a numerical study of the trade-off effects between finite-time ruin probabilities and expected total discounted dividend payments is investigated under a variety of scenarios. Finally, a generalized threshold-based strategy with a delayed dividend payment rule is studied under the compound binomial model.
MSC:
60G50 Sums of independent random variables; random walks
60K05 Renewal theory
91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. Akahori, Some formulae for a new type of path-dependent option, Annals of Appled Probability, 5, 383, (1995) · Zbl 0834.90026
[2] S. Asmussen; F. Avram; M. Usabel, Erlangian approximations for finite-horizon ruin probabilities, ASTIN Bulletin, 32, 267, (2002) · Zbl 1081.60028
[3] A. S. Alfa; S. Drekic, Algorithmic analysis of the sparre Andersen model in discrete time, ASTIN Bulletin, 37, 293, (2007) · Zbl 1154.62076
[4] B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13, 217, (2009)
[5] B. Bao, A note on the compound binomial model with randomized dividend strategy, Applied Mathematics and Computation, 194, 276, (2007) · Zbl 1193.91062
[6] S. Cheng; H. U. Gerber; E. S. W. Shiu, Discounted probabilities and ruin theory in the compound binomial model, Insurance: Mathematics and Economics, 26, 239, (2000) · Zbl 1013.91063
[7] M. Chesney; M. Jeanblanc-Picqué; M. Yor, Brownian excursions and Parisian barrier options, Advances in Applied Probability, 29, 165, (1997) · Zbl 0882.60042
[8] E. C. K. Cheung; J. T. Y. Wong, On the dual risk model with Parisian implementation delays in dividend payments, European Journal of Operational Research, 257, 159, (2017) · Zbl 1394.91204
[9] H. Cossette; D. Landriault; E. Marceau, Ruin probabilities in the discrete time renewal risk model, Insurance: Mathematics and Economics, 38, 309, (2006) · Zbl 1090.60076
[10] I. Czarna; Z. Palmowski, Ruin probability with Parisian delay for a spectrally negative Lévy process, Journal of Applied Probability, 48, 984, (2011) · Zbl 1232.60036
[11] I. Czarna; Z. Palmowski, Dividend problem with Parisian delay for a spectrally negative Lévy process, Journal of Optimization Theory and Applications, 161, 239, (2014) · Zbl 1296.91150
[12] I. Czarna; Z. Palmowski; P. Świątek, Discrete time ruin probability with Parisian delay, Scandinavian Actuarial Journal, 2017, 854, (2017) · Zbl 1402.91188
[13] A. Dassios, The distribution of the quantile of a Brownian motion with drift and the pricing of related path-dependent options, Annals of Appled Probability, 5, 389, (1995) · Zbl 0837.60076
[14] A. Dassios; S. Wu, On barrier strategy dividends with Parisian implementation delay for classical surplus processes, Insurance: Mathematics and Economics, 45, 195, (2009) · Zbl 1231.91430
[15] B. de Finetti, Su un’impostazione alternativa Della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2, 433, (1957)
[16] D. C. M. Dickson, Some comments on the compound binomial model, ASTIN Bulletin, 24, 33, (1994)
[17] D. C. M. Dickson; H. R. Water, Some optimal dividends problems, ASTIN Bulletin, 34, 49, (2004) · Zbl 1097.91040
[18] S. Drekic; A. M. Mera, Ruin analysis of a threshold strategy in a discrete-time sparre Andersen model, Methodology and Computing in Applied Probability, 13, 723, (2011) · Zbl 1245.91043
[19] H. U. Gerber, Mathematical fun with compound binomial process, ASTIN Bulletin, 18, 161, (1988)
[20] H. U. Gerber; E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2, 48, (1998) · Zbl 1081.60550
[21] S. S. Kim; S. Drekic, Ruin analysis of a discrete-time dependent sparre Andersen model with external financial activities and randomized dividends, Risks, 4, p2, (2016)
[22] B. Kim; H.-S. Kim; J. Kim, A risk model with paying dividends and random environment, Insurance: Mathematics and Economics, 42, 717, (2008) · Zbl 1152.91589
[23] D. Landriault, Randomized dividends in the compound binomial model with a general premium rate, Scandinavian Actuarial Journal, 2008, 1, (2008) · Zbl 1164.91032
[24] D. Landriault; J.-F. Renaud; X. Zhou, An insurance risk model with Parisian implementation delays, Methodology and Computing in Applied Probability, 16, 583, (2014) · Zbl 1319.60098
[25] M. A. Lkabous; I. Czarna; J.-F. Renaud, Parisian ruin for a refracted Lévy process, Insurance: Mathematics and Economics, 74, 153, (2017) · Zbl 1394.60046
[26] S. Li, On a class of discrete time renewal risk models, Scandinavian Actuarial Journal, 2005, 241, (2005) · Zbl 1142.91043
[27] S. Li, Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time renewal risk models, Scandinavian Actuarial Journal, 2005, 271, (2005) · Zbl 1143.91033
[28] R. Loeffen; I. Czarna; Z. Palmowski, Parisian ruin probability for spectrally negative Lévy process, Bernoulli, 19, 599, (2013) · Zbl 1267.60054
[29] K. P. Pavlova; G. E. Willmot, The discrete stationary renewal risk model and the gerber-shiu discounted penalty function, Insurance: Mathematics and Economics, 35, 267, (2004) · Zbl 1081.60028
[30] A. Pechtl, Some applications of occupation times of Brownian motion with drift in mathematical finance, Journal of Applied Mathematics and Decision Sciences, 3, 63, (1999) · Zbl 0933.91017
[31] E. S. W. Shiu, The probability of eventual ruin in the compound binomial model, ASTIN Bulletin, 19, 179, (1989)
[32] D. W. Sommer, The impact of firm risk on property-liability insurance prices, Journal of Risk and Insurance, 63, 501, (1996)
[33] J. Tan; X. Yang, The compound binomial model with randomized decisions on paying dividends, Insurance: Mathematics and Economics, 39, 1, (2006) · Zbl 1147.91349
[34] G. Venter; A. Underwood, Value of risk reduction, Casualty Actuary Society E-Forum, 2, 1, (2012)
[35] G. E. Willmot, Ruin probabilities in the compound binomial model, Insurance: Mathematics and Economics, 12, 133, (1993) · Zbl 0778.62099
[36] J. T. Y. Wong; E. C. K. Cheung, On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps, Insurance: Mathematics and Economics, 65, 280, (2015) · Zbl 1348.91189
[37] J.-K. Woo, A generalized penalty function for a class of discrete renewal processes, Scandinavian Actuarial Journal, 2012, 130, (2012) · Zbl 1277.60146
[38] X. Wu; S. Li, On the discounted penalty function in a discrete time renewal risk model with general interclaim times, Scandinavian Actuarial Journal, 2009, 281, (2009) · Zbl 1224.91094
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.