A Dual Risk Model With Parisian Delay Dividends

In the ccontext of ruin theory, most judgements are made on a direct sense. An example is the determination of ruin, in which a business is declared ruin when the surplus attains a negative surplus. Another example is the decision on dividend payment, in which the business pays dividends whenever the surplus level overshoots a certain threshold. From a practical view, such a scheme of making decision is generally unrealistic. Therefore,the Parisian concept is invoked to handle this issue. This idea is deemed more realistic since it allows certain delay in the decisions. Parisian concept is generally utilized in two different aspects. The first one is to incorporate this concept on the definition of ruin time, and give Parisian ruin time. Under such a setting, a business is considered ruined only when the surplus continuously stays negative for a prescribed length of time. The second contribution is to incorporate this concept of the decision for dividend payment.Specifically, a business only pays dividends when the surplus stays above certain threshold continuously for a prescribed length of time. The main problems that we discussed in the paper are the first Parisian dividend time of the dual risk model and the probability of the Parisian dividend before ruin in the dual risk model.The first chapter is divided into three parts. The first part mainly introduces the development history of the dual risk model and the Parisian concept. In the second part,we introduce the classical risk model, and introduce the dual risk model in the last part.In the second chapter, we analyze the connection between the classical risk model and the dual risk model and give the Laplace transform of the first Parisian dividend time through the Laplace transform of the Parisian ruin time of the classical risk model.The third chapter is divided into two parts and is studied the probability of the Parisian dividend before ruin in the dual risk model from two point of views respectively.In the first part, we consider this problem from the opposite side, and give the definition of hd(u). It is not difficult to find that, when r = 1, ? = 0, hd(u) is the probability of ruin before the Parisian dividend in the dual risk model with the initial value of u. And then 1 - hd(u) is the probability of the Parisian dividend before ruin in the dual risk model.And we give the intergo-differential equation of hd(u) and its solutions with exponential jumps or the mixture exponential jumps. In the second part, we give the definition of V?(u;b) and give the intergo-differential equation of V?(u;b). Its solutions with exponen-tial jumps or the mixture exponential jumps are also obtained in this part.