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The Markov additive risk process under an Erlangized dividend barrier strategy. (English) Zbl 1338.91081
Summary: In this paper, we consider a Markov additive insurance risk process under a randomized dividend strategy in the spirit of H. Albrecher et al. [Astin Bull. 41, No. 2, 645–672 (2011; Zbl 1239.91072)]. Decisions on whether to pay dividends are only made at a sequence of dividend decision time points whose intervals are Erlang(\(n\)) distributed. At a dividend decision time, if the surplus level is larger than a predetermined dividend barrier, then the excess is paid as a dividend as long as ruin has not occurred. In contrast to Albrecher et al. [loc. cit.], it is assumed that the event of ruin is monitored continuously [B. Avanzi et al., Insur. Math. Econ. 52, No. 1, 98–113 (2013; Zbl 1291.91088); Z. Zhang, J. Ind. Manag. Optim. 10, No. 4, 1041–1058 (2014; Zbl 1282.91164)], i.e. the surplus process is stopped immediately once it drops below zero. The quantities of our interest include the Gerber-Shiu expected discounted penalty function and the expected present value of dividends paid until ruin. Solutions are derived with the use of Markov renewal equations. Numerical examples are given, and the optimal dividend barrier is identified in some cases.

MSC:
91B30 Risk theory, insurance (MSC2010)
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
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