The Premium Collection And Collection Interval Are FGM-copula's Two Types Of Risk Models

In the field of random process and its applications,articles about risk models are numberless.People are not merely content to study the classical risk model.As of now. renewal risk models,risk models with diffusion,models whose claim process is dependent, models with random premium and dual models have been studied widely.A new kind of dependent model is considered in this paper,which is a further promotion for the premium process in the basis of the models with random premium.We assume that the premium and the inter-premium arrivals are dependent.The dependent structure is established through FGM-copula function.Claim process is still a compound Poisson process.This paper mainly consider two kinds of risk models.The first is the direct expansion of models with random premium.The second is the further extension of the first kind,which is the risk process with jump-diffusion.Reseatrch on these two kinds of risk models is of great significance in the insurance practice.This article is divided into three parts.The first part introduces two types of risk models and their relevant backgrounds. The first kind of model is: The second is:The second part discusses two types of dividend strategies of the first risk model: barrier and threshold.we obtain the equation of the expected discounted penalty function mb,δ(u;b):(λ1+λ2+δ)mb,δ(u;b)=λ2∫u0mb,δ(u-y;b)dF(y+λ2)∫∞uw(u,y-u)dF(y)+∫b-u0mb,δ(u+x;b)gx,w(x,0)dx+∫∞b-umb,δ(b;b)gx,w(x,0)dx. Integral equation for the expected disounted dividend payments until ruin V(u;b)is:(λ1+λ2+δ)V(u;b)=μ2∫u0V(u-y;b)dF(y)+∫b-u0V(u+x;b)gx,w(x,0)dx+∫∞b-u(u+x-b+V(b;b))gx,w(x,0)dx. We do some related discussion for exponential premium and exponential claims. Equa-tions as well as specific expressions of the laplace transform and the expected discounted dividends are derived. On this basis, we can obtain the specific expressions of some rel-evant variables on the condition that the premium and the inter-premium arrivals are dependent. Results we get are consistent with earlier researches. Under a threshold s-trategy, the expected discounted dividend payments before ruin satisfies the following integro-differential equation: When0≤u≤b,(λ1+λ2+δ)V1(u;b)=λ2∫V1(u-y;b)dF(y+∫b-u0V1(u,x-b)gx,w(x,0)dx+∫∞b-uV2(u+x;b)gx,w(x,0)dx. When u>b,(12+λ2+δ)V2(u;b)=α-αV’2(u;b)+λ2∫u-b0V2(u-y;b)dF(y)+λ2∫uu-bV1(u-y;b)dF(y)+∫∞0V2(u+x;b)gx,w(x,0)dx Results are given in the form of inference for exponential premium and exponential claims in this section.The third part mainly discusses the second risk model. Similarly, we can find the integro-differential equation of the expected total discounted dividends until ruin under the barrier strategy:(λ1+λ2+δ)Vσ(u;b)=1/52σ2V"σ(u;b)+λ2∫u0Vσ(u-y;b)dF(y)+∫b-u0Vσ+∫∞b-u(u+x-b+Vσ(b;b))gx,w(x,0)dx.The expected total discounted dividends until ruin under the threshold strategy satisfies the integro-differential equation: when0<u<b, when u> b, Besides, this article also have some discussion in detail in the particular case which is similar to the second part.This paper mainly discusses the concrete expressions of dividend function in special circumstances. For more general cases, it remains to be solved.