Research On Dividends And Ruin Problems In The MAP-modulated Risk Model

Insurance companies usually face the influence of various random factors,the most common random factors are the randomness of insurance market and the financial market.So in this paper,the random behavior of insurance com-panies is introduced into the risk model in consider of the market stochastic factors.It is generally assumed that the assets of insurance companies are ob-served in continuous time,but this assumption is difficult to realize in reality.Because the assets accounting and dividends of insurance companies usually occur at discrete time points,while these discrete points are random.For de-picting the market random factors and the random behavior at the same time,this article introduce the MAP(Markov arrival proccess):the randomness of the market environment is described by the phase process of the MAP and the ran-domness of the observation behavior of insurance companies is characterized by the jump of the MAP.That is,the insurance company's surplus process is prepared by the phase process of the MAP,the insurance company can observe the company's funds on some of the jump points of the MAP and can't observe the company's funds on other jump points,so that the model is more in line with the actual situation.The main content of this paper can be dividend into two parts:The fist part,the cdividend problem of the MAP-modulated general risk model is discussed under the threshold dividend strategy.At the moment,the bankruptcy problem of the MAP-modulated general risk model is consid-ered under no dividend strategy.The integral-differential equations satisfied by the expected discounted dividend and the expected discounted penalty(Gerber-sliiu)function and the bouncdary ccondit.ions are obtainecd.Because of the com plexity of these integro-differential equations,they are difficult to find out the corresponding display solution,so the numerical solution of these integro-differential equations care obtained by using the Sine numerical algorithm.Thro-ugh numerical examples,the influences of different market environment on the dividend amount and the probability of bankruptcy under the MAP-modulated general risk model are discussed.The second part,the dividend problem of the MAP-nmodulated dual risk model is discussed under the threshold dividend strategy.At the moment,the bankruptcy problem of the MAP-modulated dual risk model is considered un-der no dividend strategy.The integral-differential equations satisfied by the expected discounted dividend and the expected discounted penalty(Gerber-shiu)function and the boundary conditions are obtained.Since the MAP-modulated dual risk model is similar to the MAP-modulated general risk model in form,the integral-differential equations are similar,so the numerical analysis and the numerical examples are no longer considered.