The Research Of Ruin Problems In The Discrete-time Compound Renewal Risk Model With Dependence

The insurance industry is an unique risk management belong financial services industry. To any insurance company, the biggest risk is a lack of solvency. As an important component of modern risk theory, ruin theory can forecast solvency of insurance company , it is also the core of actuarial mathematics. It uses random process, advanced mathematical tools, based on the assumption fitting reality.By establishing and analysing models, we can get the earning process of the insurances and then calculate the probability of insurers'insolvency, ruin probability ,surplus before ruin and so on.The study of ruin problems can not be separated from risk models. Ordinary renewal risk model is built on the assumption that there is only one customer premise a claim at the same time, while insurance companies often come across have more than one customer claimed at the same time in car insurance and catastrophic insurance. So we will extend to study compound renewal risk model. General compound renewal risk model is based on assumption of independence between the interclaim r.v. and the claim amount r.v.. This is inconsistent with the reality of many situations clearly. For example, the strength of a catastrophe and the time elapsed since the last catastrophe are associated in catastrophic events; besides it is often difficult to achieve continuous observation of the time in reality ,so premium charge is discrete, such as Annual , quarter, month, and so on. Therefore, in order to make compound renewal risk model is more realistic, the paper added both discrete and dependent conditions in the model, soughting to study ruin problems under a more realistic risk model - discrete compound renewal risk model with dependence. And we hope to provide the early warning of financial system of insurance company and theoretical support of the management of the China Insurance Regulatory Commission.This paper is based on domestic and foreign literatures. We regulated adjustment coefficients and the finite time ruin probability under several special conditions. By numerical examples, we verify relevant conclusions of adjustment coefficient with Maple under special cases in the discrete dependent compound renewal risk model. For the finite time ruin probability of this model, obtained the probability values with different initial values under special cases by Matlab software and then we analysis the influence of the dependence between claim amounts and claim intervals to the finite time ruin probabilities. This article is divided into five chapters.The first chapter is about the meaning of studying the ruin probability and the ruin probability models. Then we pointed out the weakness of classical risk model and the risk model of the paper is more close to reality.And we overview abroad and domestic studies of this model. Lastly we put forward innovations and shortcomings of this article.In the second chapter we introduce of development of the risk theory and the composition of the general risk model, and then introduce classical risk model——L-C classical ruin probability model, and then expand to the discussion of the discrete model and the renewal risk model based on the former risk model and the ruin problems of the three models. We also review literatures the works of these three models to clear the obstacles for the more complex model in the next chapter.Because the model studied in this paper is built under dependencies, mainly using Copula functions as a tool.Therefore ,we first introduce some of the theories on the Copula before introducing the model subsequently. This chapter introduces the discrete compound renewal risk model with dependence and discusses the general impact of dependence on the adjustment coefficient. Dependence here is about claim amounts and the interclaim. In certain exceptional cases we shows the impact of the relevance the adjustment coefficient in this model by the numerical example finally. Specific for: For ( X , W )which has a bivariate geometric distribution which is defined with Frank Copula and the FMG Copula.And we testify the impact of dependence on the adjustment coefficient in the model with Maple software. Then We get:ρ( ?5 0)≤ρ( ?2 0)≤ρ( ?10)≤ρ(10)≤ρ(20)Under the FMG Copula; and under Frank Copulaρ( ?5 0)≤ρ( ?2 0)≤ρ( ?10)≤ρ(0)≤ρ(10)≤ρ(20) . This conclusion is the same as the ones under continuous line risk model.In chapter four this paper we establish dependence with geometric distribution which is defined with Frank Copula and FMG Copula as the background, carrying out numerical simulation in this model, using Matlab software.The finite time ruin probability value is obtained under different initial values of u and different correlation under some special cases. To some extent, we can show the imapact of the dependence between interval time r.v. and claim amount r.v. on the ruin probability .The conclusion is the same as the one in the compound continuous renewal risk model under the FMG Copula.The last chapter summarized the conclusions of special case of numerical examples : First, we can still get ( X , W )≤co( X′, W′)whenρ≤ρ′under the model in the paper; second, when the dependence of ( X , W ) is with geometric distribution which is defined with Frank Copula and FMG Copula we obtained for all the initial surplus u, dependencies and time ruin probability is proportional. Then we analyzed the reasons for the results.