Insurance Actuarial Risk Models In Markovian Environment Or Copula Dependence

This dissertation focuses on the ruin problems in the specific situation of stochastic environment and Copula dependence by applying the mathematical tools such as Markovian process, stochastic analysis, renewal measure, and Cop-ulas. The main contents include the following aspects.Firstly, the author proposes the application of the basic structure of the classical risk model and makes a clarification for both the study of the compound Poisson and the compound binomial models. The possible developments for the classical risk model are also clarified in order to draw out the discussion in the later chapters.Secondly, the author suggests to put the compound binomial model into the stochastic environment which is created by the Markovian process. With the employing the method of up-crossing zero points-renewal measure, the expression of the renewal mass function and the recursive formula of the joint distributions of the ruin time, the surplus before ruin and the deficit at ruin will be provided successively. Meanwhile the supremum distribution for the surplus is also put into consideration with the same strategy. Furthermore, a simple example is provided to explain the feasibility of the conclusion which is drawn from the above part.Thirdly, the author considers the two-dimensional compound binomial model in Markovian environment, and defines three types of ruin time, and the corre-sponding finite-time ruin probabilities. Then the recursive formula of the finite-time survival or ruin probabilities are derived separately for those three types. Meanwhile, some results about the specific deficits are also taken into considera-tion. Then, the allocation for the initial capital is considered using the finite-time survival probabilities.Fourthly, during the process of applying the Copulas in the two-dimensional risk models, with the consideration of the dependence relation between the claim amounts, the author chooses the FGM Copula to be put into application. For the compound Poisson model, a partial integro-differential equation satisfied by the survival probability is derived. The recursive formula of the finite-time survival or ruin probabilities are then derived separately in the situation of continuous and discrete distribution of claim amount, for the compound binomial model. Furthermore, the uncertainty for the Copulas under the discrete case is also illustrated.