The Dividend Problem Of Several Kinds Of Risk Models

In recent years, the issue of dividends has received remarkable attention in the actuarialmathematics. In this paper, we consider the dividends of several types of risk models. Therisk models including the one-dimensional difusion risk model, the classical risk model andthe classical risk models perturbed by difusion; dividend strategies mainly involve barrierdividend strategies, threshold dividend strategies and hybrid dividend strategies; the prob-lems are studied including exit times, expected discounted dividends until ruin, the momentsand moment-generating functions of discounted dividends until ruin, Gerber-Shiu functionand Laplace transform of the ruin time; the methods contain exit time, special function,Markov process, Taylor’s formula, Dynkin’s formula, and so on. We obtain the detailedresults for some special problems. According to the contents, the article can be divided intothree chapters:1) The exit time and the dividend value function for one-dimensional difusion processes.In this chapter, we investigate the exit times from an interval for a general one-dimensionaltime-homogeneous difusion process and their applications to the dividend problem in risktheory. Specifcally, we frst use Dynkin’s formula to derive the ordinary diferential equa-tions satisfed by the Laplace transform of the exit times. Then, as example, we get theclosed-form expressions of the Laplace transform of the exit times for several popular d-ifusions, which are commonly used in modelling of fnance and insurance market. Mostinteresting, as the applications of the exit times, we create the connect between the dividendvalue function and the Laplace transform of the exit times. Both the barrier and thresholddividend value functions are clearly expressed in terms of the Laplace transform of the exittimes.2) The classical risk model with a hybrid dividend strategy.In this chapter, we consider the classical risk model with a hybrid dividend strategy.We frst expatiate the classical risk model, the hybrid dividend strategies and the researchbackground of the dividend strategies problems. Then under the classical risk model witha hybrid dividend strategy, we defne expected discounted dividend until ruin, the momentsand moment-generating functions of discounted dividend until ruin, Gerber-Shiu functionand the Laplace transform of the ruin time. Next, we derive the integro-diferential equa-tions with boundary conditions satisfed by the expected discounted dividends until ruin.We also obtain integro-diferential equations with boundary conditions for the moments and moment-generating functions of discounted dividends until ruin, respectively. The Gerber-Shiu function and Laplace transform of the ruin time are also discussed. The special casethat the claim size distribution is exponential is considered in some details.3) The classical risk model disturbed by difusion with a hybrid dividend.Based on the previous chapter, we investigate the compound Poisson risk model per-turbed by difusion with a hybrid dividend strategy in this chapter. We continue to usethe symbols in the previous chapter for convenience. The denote of ruin time will difer-ent result from the diferent risk model. We frst introduce the classical risk model, andpoint out that there are diferent from Chapter2because of the change of the risk mod-el. We then use Dynkin’s formula to derive the integro-diferential equations and boundaryconditions satisfed by the expected discounted dividends until ruin, and, as example, weobtain the explicit expression for the expected discounted dividends until ruin by applyingthe integro-diferential equations and boundary conditions when claims are exponentiallydistributed. We also derive integro-diferential equations with boundary conditions for themoment-generating functions and kth moments of discounted dividends until ruin. Finally,the famous Gerber-Shiu function is discussed, and the closed-form expression for the Laplacetransform of the ruin time is solved when claims are exponentially distributed. In addition,we cite the results in Chapter2when prove the boundary conditions.