Applications Of Stochastic Process And Stochastic Control Theory In Risk Theory

Insurance mathematics is an applied mathematics used to study the mathematical basis of insurance institutions. It is concerning the value of premium and compulsory capital, reinsurance system, emergency capital, profit analysis and risk theory, etc. Risk theory is the most theoretical part in insurance mathematics. It has been concerned more than one hundred years. In risk theory, various kinds of random model derived from insurance business are considered.With no difference from the managers of other kinds of company, the insurers have two purposes:minimizing the risk and maximizing the profit. Risk theory just develops according to these two purposes. On the one hand, the risk of an insurance company can be depicted by some actuarial variables, for example, the probability of ruin, the time of ruin, the surplus immediately prior to ruin, the deficit at ruin and so on. The initial study of risk theory mainly focused on these important actuarial variables. In order to minimize the risk, the insurers will take the appropriate measures such as reinsurance, investment in finance market. On the other hand, at the same time of minimizing the risk, the insurers care about their profit. The total dividends paid before the time of ruin is the most representative profit for the insurers or shareholders. The so-called dividends are the portion of surplus that paid to shareholders or providers of the initial capital.My dissertation focus on the problems stated above, and more definitely, joint distributions of some actuarial random vectors in the compound binomial model, some measures of the severity of ruin in the compound binomial model, the optimal dividend problem and the voluntary ruin barrier in the compound Poisson model with debit interest, optimal dividend control with dynamic proportional reinsurance policy in the bivariate compound Poisson model. It is organized as follows:The first chapter is introduction. In this chapter, the risk models discussed in my dissertation—the compound binomial model, the compound Poisson model with debit interest and the bivariate compound Poisson model—are introduced briefly. And the background and present development of relevant problems are sketched.In the second chapter, joint distributions of some actuarial random vectors in the compound binomial model are studied. A renewal mass function of a defective renewal sequence constituted by the up-crossing zero points of the model is introduced. And the explicit expression is obtained. By the mass function together with Markov property of the surplus process, the explicit expressions of the ruin probability, the joint distributions of some actuarial random vector (including the time of ruin, the surplus immediately prior to ruin, the deficit at ruin, the maximal deficit from ruin to recovery and the maximal deficit in the whole) are given.Seeing that deficit doesn't definitely lead to the decision to stop activities for insurers. The possibility of recovery depends on the state of the company at ruin time, but also on the claims this company could endure after that time and time spent with a negative surplus. These measures are rational basis for the company to make the decision to stop activities or not. In the third chapter, the maximal severity, the cost of recovery, the total duration, and the total cost of negative surplus in the compound binomial model are studied. Finally, an example with geometrically distributed claims is considered and all the measures of the severity of ruin, mentioned above, are calculated explicitly in this case.In the forth chapter, the optimal dividend problem and a new ruin barrier—voluntary ruin barrier in the compound Poisson model with debit interest are considered. As explained by Dickson and Waters (2004), the shareholders should be liable to cover the deficit at ruin. Therefore, the optimal dividend problem is to maximize the expectation of the difference between the accumulated discounted dividends until ruin and the discounted deficit at ruin. Seeing that the value at absolute ruin barrier is negative, it is necessary to find a new ruin barrier—voluntary ruin barrier below which the insurers would rather stop their activities than continue taking the debit. Explicit solutions are given out when the claim amount distribution is exponential. With the development of risk theory, one-dimension risk models could no longer characterize the more and more complex situation of insurance. For example, in a traffic accident, claims may come from person injury, and may come from automobile damage also. Therefore, multi-dimension dependent risk models emerged as the time require. In the fifth chapter, by dynamic control theory, the optimal dividend strategy and the optimal dynamic proportional reinsurance strategy are find out to maximize the cumulative expected discounted dividends in the bivariate compound Poisson model.