Research On Ruin Probabilities And Its' Asymptotic Relationships In Several Classes Risk Model

In risk theory in the present study most of scholars focus exclusively on generalization to the following three risk models, i.e. the compound binomial risk model, the compound Poisson risk model and renewal risk model. For example, the above three risk models may be generalized in following ways: (a) The occurrence of the claims may be described by a more general point process than the binomial process, or the Poisson process or renewal process. (b) The premiums may depend on the result of the risk business. (c) Inflation and interest may be included in the model. Then they research the ruin probability or Gerber-Shiu discounted penalty expectation function so on. But there are still many scholars who study the above three risk models. We think that their woks are necessary and significance very much. Because although there are about one hundred years since people started to study these risk models and obtained many classical results, there are still many problemsh are not solved. For example the exprise formula of ruin probability in compound binomial risk model and many problems in Poisson risk model under the adjustment coefficient does not exist are all not solved too. In a word all works are very important and meaning.In these paper we not only research these risk models but also generalized these model then we studied the ruin probability, the survival probability in finite time and Gerber-Shiu discount expectation function, i.e. m(x)=E{v^TW(R(T-),|R(T)|)1_{(T＜∞)}|R(O)=x}, in our generalized risk model and obtained there express or asymptotic relationship.In the fully discrete compound binomial risk model according to the Markov property of the model we firstly obtained the joint density distribution of the series of the inter-claim times, i.e.{T_i,i = 1,2,3,...}and the surpluses of insurance company at claim time, i.e. {R(U_i),i = 1,2,3,...}, that is where x₀ = x is the initial capital of insurance company. Then we obtained the ruin probability and the survival probability in finite timeIn the mean while we obtained the probability that the insurance company has survived to time t and its surplus is more than Munder the initial surplus is u, i.e.where R_j =x+C(k₁ +k₂ +…+k_j)--(i₁ +i₂ +…+i_j-1),j=1,2,3,…,n·Q(n,m)m{(k₁,k₂,...,k_n):k₁ +k₂ +...+k_n =m, k_i∈Z⁺}.A(M, k, t)≡R_k - (M + k - t)1_{{M+k-t≥0}}·On the other hand if the units of money per unit time c=1, we firstly obtained the delayed renewal equation of ruin probability based on the Markov property of the risk process. Then by probability generating function we obtained the Pollazek-Khinchin formula:About the asymptotic relationship of the ruin probability or Gerber-Shiu discount expectation function, we obtained the delayed renewal equation of Gerber-Shiu discount expectation function according to the Markov property of the process and by the method of probability generating function and renewal theory, i.e.(2)θ=θ(v)is the only positive root of equation vpf(z)=z-vq about variable z in interval (vq, v]. is a probability generating function,here F(j) = f(j + 1)+ f(j + 2)+ f(j + 3)+ ..., denotes the tail distribution of F.(4) Ifv＜1, then Q(v-θ)=(1-θ)/(1-θ),Ifv=1, then Q(v)=pμ. Then based on the above results and under the adjustment coefficient R exists we obtained the asymptotic relationship of Gerber-Shiu discount expectation functionIn the general compound binomial risk model there is less good result because we can not use the method of generating function as in fully discrete compound binomial risk model. We firstly obtained the joint density distribution of the series of the inter-claim times, i.e.{T_i,i = 1,2,3,...} and the surpluses of insurance company at claim time, i.e.{R(U_i),i=1,2,3,...}, by applying the same mean as in fully discrete compound binomial risk model. That is P_X(T₁ = k₁,R(U₁)∈dx₁,...,T_n = k_n,R(U_n)∈dk_n)=g_n {x,k₁,...,k_n,x₁,...,x_n}dx₁ ...dx_n where then based on the above result we obtained the ruin probability formula and the survival probability in finite timeIn Poisson risk model we focus exclusively on the following ways. One is the problem of the asymptotic relationship of ruin probability or its local result when the claim distribution function belongs to. Other is to generalize the Poisson risk model and study the ruin probability and the Gerber-Shiu discounted penalty expectation function.The distribution class S^*(v) is a class of between light-tailed distribution and heavy-tailed distribution. When v=0, it belongs to light-tailed distribution class but when当v＞0 it belongs to heavy-tailed distribution class. In Poisson risk model if claim distribution function belongs to heavy-tailed distribution class then it is obvious that the adjustment coefficient does not exist. If the claim distribution function belongs to the light-tailed distribution class, then the adjustment coefficient may exist or not. In the present study people generally research the risk model under the adjustment coefficient do exist. In this paper we studied the Poisson risk model under not only the claim distribution function belongs to S^*(v) but also the adjustment coefficient does not exist and obtained the asymptotic relationship of ruin probability and the local result of ruin probabilityAbout the problem of the model generation we mainly focus on the following three aspects as described in paragraph one. But in this paper we only considered one condition of generalization (b), that is, the premiums is described by a Poisson process and obtained the ruin probability formula under the adjustment coefficient does exist then based on the formula we obtained the results when the claim distribution function is exponentially distributed. And if we assume the initial surplus is non-negative integer and the claim is positive integer we obtained the ruin probability formulaAbout generalization (a) we respectively considered the condition that only one risk accident occurs in a very small time internal but at this time there may be more than one claim to occurs and meanwhile the insurance company used non-claim discount system. In this paper we firstly discussed the condition that the claim number is a Poisson process under the risk accident occurs. Where the generalized model is called the compound generalized Poisson risk model. We obtained the Pollazek-Khinchin formula of ruin probability by transforming the model to classic Poisson. Then we gave the upper and down boundary based on the above formula when the claim distribution function is exponentially distributed. Secondly we further considered the compound generalized Poisson risk model with non-claim discount system and introduced a new risk model, which the claim point process is described by a compound process that is composed by a Poisson process and defective geometric process, This model is called the compound compound Poisson defective geometric model. In this model we obtained the defective renewal equation of Gerber-Shiu discounted penalty expectation function and ruin probability. Thirdly we studied the compound Poisson-Geometric risk model which is introduced by Mao Zechun and Liu Jine(2005) and researched the same problems and further obtained the Pollazek-Khinchin formula of ruin probability. In classic Poisson risk model we considered the condition under not only with non-claim discount system but also the premiums depending on the non-claim discount and gave the method of determine the valve of non-claim discount. In classic Poisson risk model perturbed by diffusion when claim distribution function F∈S^*, we obtained the local result of survival probability holds for all z＞0.Finally we studied the delayed renewal risk model under the claim distribution function F∈S^* and obtained the same local result of survival probability as (*).