The Research On Ruin Theory And Dividend Strategy For Insureance Risk Model

Risk theory plays an important role in financial mathematics and actuary, it through the study of stochastic risk model in the insurance industry to deal with several actuarial variables, such as the ruin probability, ruin time, deficit at ruin, surplus immediately prior to ruin, Gerber-Shiu expected discounted penalty function, expected discounted dividend function, adjustment coefficient, etc. Early research on insurance risk model can be traced back to the results of Lundberg(1903). It was because of his work, which lay a solid foundation for the insurance risk theory. Until today there are a large number of related papers and monographs, which generalize the work of Lundberg(1903) and in-depth study, such as the perturbed risk model to come, renewal risk model, compound binomial risk model, absolute ruin risk model, Markov regime-switching risk model and dependent risk model, etc.In addition, the dividend strategy risk model are also received widespread attention, which is inseparable with the realistic significance of dividends. Dividends mean the in-surance company pays certain surplus to the shareholders or the initial reserve provider. The dividend amount also reflects a company’s economic efficiency and strength. The dividend strategy was first discussed by De Finitti(1957) at the15th International Congress of Actuaries in1957. He pointed out that the company should maximize the expected discounted dividends before ruin. The current common dividend strategies are barrier dividend strategy, threshold dividend strategy, band dividend strategy, linear dividend strategy, etc.On the basis of these background, my doctoral dissertation will be devoted to doing some researches in the following aspects:Firstly, I will make the insurance risk model and problem more practical. Secondly, according to the characteristics of the current risk model and problem, giving full play to the role of the theory of stochastic process, I will try to find the way to solve the problem. Finally, in order to make the research results have a very good guide to practice, I will try my best to give the explicit expressions or numerical examples. In the following, I will introduce the content of every Chapter.Chapter1. We introduce several insurance risk models and confluent hypergeo-metric equation, etc.Chapter2. We consider the absolute ruin risk model with credit interest under the threshold dividend strategy, and obtain the integro-differential equations with boundary conditions satisfied by the moment-generating function and n-th moment of the present value of all dividends until absolute ruin, Gerber-Shiu expected discounted penalty func-tion, the Laplace transform of the first time to reach the dividend barrier. When the claim sizes have a exponential distribution, we get the explicit expressions for the n-th moment of the present value of all dividends until absolute ruin and the Laplace trans-form of absolute ruin time. In particular, in the case of n=1we provide the numerical examples and illustrate the impacts of threshold b, discount interest force, credit interest and debit interest on the expected discounted dividend function.This chapter is mainly based on the paper:Yu Wenguang, Huang Yujuan. On the time value of absolute ruin for a risk model with credit and debit interest under a threshold strategy. Science China Mathematics, under review.Chapter3. We study the absolute ruin problems for the perturbed compound poisson risk model with credit interest under the threshold dividend strategy, and get the integro-differential equations with boundary conditions satisfied by the moment-generating function and n-th moment of the present value of all dividends until absolute ruin, Gerber-Shiu expected discounted penalty function. When the claim sizes follow exponential distribution, we derive the explicit expressions for the n-th moment of the present value of all dividends until absolute ruin when discount interest force α=0. Specially, when n=1and α>0we provide the numerical examples and explain the impacts of threshold b, discount interest force, credit interest and debit interest on the expected discounted dividend function.This chapter is mainly based on the paper:Yu Wenguang. Some results on absolute ruin in the perturbed insurance risk model with investment and debit interests. Economic Modelling,31(2013),625-634.Chapter4. We study the absolute ruin Markov risk model under the barrier dividend strategy, and derive the integro-differential equations with boundary conditions satisfied by the moment-generating function and n-th moment of the present value of all dividends until absolute ruin and Gerber-Shiu expected discounted penalty function. In addition, we further consider a class of semi-Markovian dependent absolute ruin risk model. Under the framework, at each instant of a claim, the Markov chain jumps to a state j, and the distribution Fj(y) of the claim depends on the new state j. Then the next interarrival time is exponentially distributed with parameter A^. Note that given the states Zn-1and Zn, the quantities Wn and Xn are independent, but there is autocorrelation among consecutive claim sizes and among consecutive interclaim times as well as cross-correlation between Wn and Xn.This chapter is mainly based on the paper:Yu Wenguang, Huang Yujuan. Dividend payments and related problems in a Markov-dependent insurance risk model under absolute ruin. American Journal of In-dustrial and Business Management,1(1)(2011),1-9.Yu Wenguang, Huang Yujuan. The Markovian regime-switching risk model with constant dividend barrier under absolute ruin. Journal of Mathematical Finance,1(3)(20183-89.Chapter5. We study a discrete risk model with randomized dividends and s-tochastic premium income, where the premium income process and claim process follow compound binomial process. The insurer pays a dividend of1with a probability go when the surplus is greater than or equal to a nonnegative integer b. We derive the recursion formulas for the expected discounted penalty function. As applications, we present the recursion formulas for the ruin probability, the distribution function of the deficit at ruin and the generating function of the deficit at ruin. Finally, numerical examples are also given to illustrate the effect of the related parameters on the ruin probability.This chapter is mainly based on the paper:Yu Wenguang. Randomized dividends in a discrete insurance risk model with s-tochastic premium income. Mathematical Problems in Engineering,2013(2013),1-9.Chapter6. We consider the risk model with a dependent setting where the time between two claim occurrences determines the distribution of the next claim size. An integro-differential equation for some Gerber-Shiu expected discounted penalty function for the exponentially distributed claim sizes is derived. Applications of the integro-differential equation are given to the Laplace transform of the time of ruin, the deficit at ruin, the surplus immediately before ruin occurs. Finally, we analyze the Gerber-Shiu expected discounted penalty function and the expected discounted dividend function in the same risk model with a constant dividend barrier.This chapter is mainly based on the paper:Yu Wenguang, Huang Yujuan. Some results on a risk model with dependence between claim sizes and claim intervals.数学杂志,33(5)(2013),781-787. Chapter7. We study a Markovian regime-switching risk model (also called Markov modulated risk model) with stochastic premium income, in which the premium income process, the claim process and discount interest force process are driven by Markovian regime-switching process. The purpose of this section is to study the integral equations satisfied by the expected discounted penalty function. Applications of the integral equa-tions are given to be explicit expression of Laplace transform of the time of ruin, the deficit at ruin and the surplus immediately before ruin occurs in the case of one state and exponential distribution. Finally, numerical example is also given to illustrate the effect of the related parameters on these quantities.This chapter is mainly based on the paper:Yu Wenguang. On the expected discounted penalty function for a Markov regime switching risk model with stochastic premium income. Discrete Dynamics in Nature and Society,2013(2013),1-9.