On the dual risk model with Parisian implementation delays in dividend payments

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. Avanzi, Gerber, and Shiu (2007)) 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 Dassios and Wu (2009)). 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. Asmussen, Avram, and Usabel (2002)) 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.