Asymptotic Analysis Of Ruin Probabilities Of Bidimensional Risk Models

It is well known that ruin probability is one of the main research object of theactuarial mathematics and applied probability theory. In many cases, it is not easy tocalculate the value of ruin probability. So the asymptotic researches of ruin probabilityare particularly important, and the asymptotic estimates of ruin probability for riskmanagement have important theoretical significance. From Lundberg (1903)[69]era tothe now, the asymptotic theory of ruin probability has become a very active researcharea. At the beginning of the study, people consider the one-dimensional renewal riskmodel. But in real life, for an insurance company, it is impossible to deal in onlyone type of insurance contract. So, there is more practical significance to study avariety of insurance contracts of multidimensional renewal risk model which is put onthe agenda. But the researches of the multi-dimensional renewal risk models are oftenmore complicated than one-dimensional case and calculations are more cumbersome,and sometime it needs to solve some new mathematical problems. On the other hand,one discovered that it has no diferent between the multi-dimensional renewal riskmodels and the bidimensional renewal risk models. Therefore, in this article, we studythe uniform asymptotic theory for the finit-time ruin probabilities of bidimensionalrenewal risk model from the following three aspects.Firstly, we study the uniform asymptotics for the finite-time ruin probabilitiesof bidimensional nonstandard renewal risk model without interest rates. Two classesof claim sizes are the independent and identically distributed random variables andtheir distributions belong to the intersection of the long-tailed distribution class andthe dominatedly-varying-tailed distribution class (see, Definition1.2and1.5), andthe inter-arrival times follow an extended negatively orthant dependence structure orwidely orthant dependence structure (see, Definition1.8). When two classes of claimsshare the same arrival times, we obtain that the asymptotic formulas of ruin probabilityhold uniform for t∈[f (x),∞), where f (x) is any an infinitely increasing function. Theresults are used the diferent methods of Chen et al.(2011)[27], and expand the scope of the distribution family and dependence structure and weaken their some conditions.Secondly, we consider the uniform asymptotics for the finite-time ruin probabilitiesof two kinds of nonstandard bidimensional renewal risk models with constant interestforces and difusion generated by Brownian motions. In one of the models, two classesof claims have diferent arrival times, while in the another model, two classes of claimsshare the same arrival times. In both models, two classes of claim sizes are both uppertail asymptotically independent and their distributions belong to the intersection ofthe long-tailed distribution class and the dominatedly-varying-tailed distribution class,and the inter-arrival times follow a widely lower orthant dependence structure. Ineach model, we obtain three kinds of uniform asymptotic estimates for the finite-timeruin probabilities, respectively. These results are used the diferent methods of Li etal.(2007)[65]and Bai and Song (2011)[12]and extend the partial results of Li et al.(2007)[65]and the results of Bai and Song (2011)[12].Finally, we consider the uniform asymptotics for the finite-time ruin probabilitiesof a time-dependent bidimensional renewal risk models. In the model, two classes ofclaim sizes are the independent and identically distributed random variables and sharethe same arrival time and their distributions belong to subexponential distributionclass(see, Definition1.3). And there is a certain dependence structure between theclaim sizes and their inter-arrival times. We obtain the uniform asymptotic estimatesfor the finite-time ruin probabilities. Under the proper conditions, we extend the resultsof one-dimensional case of Asimit and Badescu (2010)[5]and Li et al.(2010)[67]to thebidimensional case. And diferent from the above two studies, we can see that thedependence structure of the claim sizes and their inter-arrival times has certain efecton the asymptotic estimation of ruin probability.The above results can be found that the nonstandard renewal risk models which aredealed with this paper have four characteristics: the dependency of random variables,the heavy-tailed property of of distributions of random variables, multi-dimensionalityof a varity of insurances and uniform of asymptotic estimates. The uniform asymptoticsshow that the sizes of the initial capital of an insurance company have nothing to dowith the length of the operation of insurance company. In other words, to control therisk, insurance company need the same size of the initial capital, no matter how manyyears it is going to run. These results not only enrich the theory of extremes valuestheory but also have potential applications in the fields of finance and insurance.