The Penalty Function, Value Function And Their Application : Dividend-Reinsurance Strategy

In this paper, we consider two risk models.In the first part of the paper, we discuss a Sparre Andersen risk model witha dividend threshold strategy. According to the definition of the Gerber-Shiu ex-pected discounted penalty function, we apply the methods of the mathematicalanalysis to derive the integro-di?erential equation and renewal equation of theGerber-Shiu expected discounted penalty function.In the second part of the paper, we discuss the discrete time risk model withdouble thresholds strategy and o?er the methods for computing the Gerber-Shiuexpected discounted penalty function and value function. On this basis, we mainlypropose a new type of reinsurance that we call dividend-reinsurance strategy. Itis introduced as follows: the reinsurer is required to pay to the insurer a givenamount of money whenever the surplus becomes negative and the deficit does notexceed the same amount of money, while in return the insurer pays to the reinsurera fraction of all the dividends as reinsurance premium. Applying the expecteddiscounted penalty function and value function, we obtain the calculation formulaof reinsurance premium and the expected present value of dividends that are paid tothe shareholders before ruin. We contrast the expected present value of dividendsdistributed by the company and ruin time in the presence of dividend-reinsurancestrategy and that in the absence of dividend-reinsurance strategy. Some resultsabout the advantages of the new strategy are proven. Under a certain condition,we find that the dividend-reinsurance strategy can not only remarkably increasethe expected present value of dividends before ruin, but also enhance the life ofthe insurance company. In this paper, we apply the contraction mapping andthe theory of the fixed point to calculate the numerical problem. The theoreticalresults are validated. The new reinsurance strategy in the paper is really worthgeneralizing.