Research On The Dependent Risks For Insurance

This thesis focuses on discussion of the effects of the dependency among the individual risks on various aspects in actuarial science, mainly including the calculation of premiums, estimation of ruin probabilities, and so on. The main contents include the following aspects.Firstly, this thesis generalizes the concept of the bivariate correlation order, proposed by Dhaene and Goovaerts (1996), to the multivariate situation, and discusses some important properties of the correlation order in the multivariate case. The relationships between the generalized correlation and some other important stochastic orders, such as supermodular order, are discussed.Secondly, this thesis combines the limit theory and method of stochastic orders to estimate the bounds of the total claim amount of insurance portfolios. The upper bounds of the total claim amount are investigated in detail for both immediate and due whole life insurance portfolios, and their limit distributions are discussed under the assumption that the individual residual lives are positively associated and negatively associated respectively.Thirdly, this thesis incorporates both copula functions and stochastic orders into theα-power approximations to estimate the survival probabilities at fractional age of multivariate life status. In doing so, this thesis proposes two new methods, FFTGS and FGSTF, for this kind of estimating. This thesis also investigates the effects of the value of the parameterαon the estimation by these two methods. A comparison through stochastic orders is also made between these two methods.Fourthly, this thesis considers a bivariate Poisson model with common shock. The asymptotic formulae for ultimate ruin probabilities are derived for claims taking several different forms of heavy tail.Finally, this thesis also derives the asymptotic formulae of both finite and ultimate ruin probabilities of a discrete time surplus model with (constant and stochastic respectively) interest rate under the general dependence structure among the individual insurance risks. The main assumption for the results is that the individual insurance risks has no bivariate upper tail dependence.