zbMATH — the first resource for mathematics

On optimal periodic dividend strategies in the dual model with diffusion. (English) Zbl 1296.91143
Summary: The dual model with diffusion is appropriate for companies with continuous expenses that are offset by stochastic and irregular gains. Examples include research-based or commission-based companies. In this context, E. Bayraktar et al. [Astin Bull. 43, No. 3, 359–372 (2013; Zbl 1283.91192)] show that a dividend barrier strategy is optimal when dividend decisions are made continuously. In practice, however, companies that are capable of issuing dividends make dividend decisions on a periodic (rather than continuous) basis.
In this paper, we consider a periodic dividend strategy with exponential inter-dividend-decision times and continuous monitoring of solvency. Assuming hyperexponential gains, we show that a periodic barrier dividend strategy is the periodic strategy that maximizes the expected present value of dividends paid until ruin. Interestingly, a ‘liquidation-at-first-opportunity’ strategy is optimal in some cases where the surplus process has a positive drift. Results are illustrated.

91B30 Risk theory, insurance (MSC2010)
93E20 Optimal stochastic control
Full Text: DOI
[1] Albrecher, H.; Bäuerle, N.; Thonhauser, S., Optimal dividend payout in random discrete time, Statist. Risk Model., 28, 3, 251-276, (2011) · Zbl 1233.91139
[2] Albrecher, H.; Cheung, E. C.K.; Thonhauser, S., Randomized observation periods for the compound Poisson risk model: dividends, Astin Bull., 41, 2, 645-672, (2011) · Zbl 1239.91072
[3] Albrecher, H.; Gerber, H. U.; Shiu, E. S.W., The optimal dividend barrier in the gamma-omega model, Eur. Actuar. J., 1, 1, 43-55, (2011) · Zbl 1219.91062
[4] Albrecher, H.; Thonhauser, S., Optimality results for dividend problems in insurance, RACSAM Rev. R. Acad. Cienc.; Ser. A, Math., 100, 2, 295-320, (2009) · Zbl 1187.93138
[5] Allen, F.; Michaely, R., (Payout Policy, Handbook of the Economics of Finance, Vol. 1A, (2003), Elsevier), 337-429, (Chapter 7)
[6] Avanzi, B., Strategies for dividend distribution: a review, N. Am. Actuar. J., 13, 2, 217-251, (2009)
[7] Avanzi, B.; Cheung, E. C.K.; Wong, B.; Woo, J.-K., On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insurance Math. Econom., 52, 1, 98-113, (2013) · Zbl 1291.91088
[8] Avanzi, B.; Gerber, H. U., Optimal dividends in the dual model with diffusion, ASTIN Bull., 38, 2, 653-667, (2008) · Zbl 1274.91463
[9] Avanzi, B.; Shen, J.; Wong, B., Optimal dividends and capital injections in the dual model with diffusion, ASTIN Bull., 41, 2, 611-644, (2011) · Zbl 1242.91089
[10] Bayraktar, E.; Kyprianou, A. E.; Yamazaki, K., On optimal dividends in the dual model, ASTIN Bull., 43, 3, 359-372, (2013) · Zbl 1283.91192
[11] Bayraktar, E.; Kyprianou, A. E.; Yamazaki, K., Optimal dividends in the dual model under transaction costs, Insurance Math. Econom., 54, 133-143, (2014) · Zbl 1294.91071
[12] Borodin, A. N.; Salminen, P., (Handbook of Brownian Motion—Facts and Formulae, Probability and its Applications, (2002), Birkhäuser Basel, Boston, Berlin) · Zbl 1012.60003
[13] Bühlmann, H., (Mathematical Methods in Risk Theory, Grundlehren der Mathematischen Wissenschaften, (1970), Springer-Verlag Berlin, Heidelberg, New York)
[14] de Finetti, B., Su un’impostazione alternativa Della teoria collettiva del rischio, (Transactions of the XVth International Congress of Actuaries, Vol. 2, (1957)), 433-443
[15] Gerber, H. U., The dilemma between dividends and safety and a generalization of the lundberg-cramér formulas, Scand. Actuar. J., 1974, 46-57, (1974) · Zbl 0281.62097
[16] Klebaner, F. C., Introduction to stochastic calculus with applications, (2005), Imperial College Press London · Zbl 1077.60001
[17] Kyprianou, A. E., Introductory lectures on fluctuations of Lévy processes with applications, (2006), Springer-Verlag Berlin · Zbl 1104.60001
[18] Morill, J. E., One-person games of economic survival, Nav. Res. Logist. Q., 13, 1, 49-69, (1966) · Zbl 0149.16904
[19] Wei, J.; Wang, R.; Yang, H., On the optimal dividend strategy in a regime-switching diffusion model, Adv. Appl. Probab., 44, 886-906, (2012) · Zbl 1251.93143
[20] Zhang, Z., Cheung, E.C.K., 2013. The Markov additive risk process under an Erlangized dividend barrier strategy. Preprint, May. · Zbl 1338.91081
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.