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On the dual risk model with Parisian implementation delays in dividend payments. (English) Zbl 1394.91204
Summary: In this paper, we study the dual compound Poisson risk process, which is suitable for a business that pays expenses at a constant rate over time and earns random amount of income at random times. In contrast to the usual dividend barrier strategy (e.g., [B. Avanzi et al., Insur. Math. Econ. 41, No. 1, 111–123 (2007; Zbl 1131.91026)]) in which any overshoot over a pre-specified barrier is paid immediately to the company’s shareholders as a dividend, it is assumed that dividend is payable only when the process has stayed above the barrier continuously for a certain amount of time $$d$$ (known as the ‘Parisian implementation delay’ in [A. Dassios and S. Wu, Insur. Math. Econ. 45, No. 2, 195–202 (2009; Zbl 1231.91430)]). Under such a modification, the Laplace transform of the time of ruin and the expected discounted dividends paid until ruin are derived. Motivated by the ‘Erlangization’ technique (e.g., [S. Asmussen et al., Astin Bull. 32, No. 2, 267–281 (2002; Zbl 1081.60028)]) of approximating a fixed time using an Erlang distribution, we also analyze the case where the delay $$d$$ is replaced by an Erlang random variable. Numerical illustrations are given to study the effect of Parisian implementation delays on ruin-related quantities and to demonstrate the good performance of Erlangization. Interestingly, unlike the traditional barrier strategy, it is found that the optimal dividend barrier maximizing the expected discounted dividends does depend on the initial surplus level.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60G51 Processes with independent increments; Lévy processes 62P05 Applications of statistics to actuarial sciences and financial mathematics
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