A Study Of The Ruin Probability In The Double Binomial Model

A Study of the Ruin Probability in the Double Binomial ModelProcessing over one hundred years history in other countries, the theory of risk model is an important contents of insurance actuarial science. In 1986, the Northern American Committee published Actuarial Mathematics, edited by Newton L. Bower, Hans U. Gerber, James C. Hickman, Donald A. Jones and Cedil J. Nesbitt in which they discussed the insurance risk model of discrete time[1]. The surplus model of the classical discrete risk model is below:, (1.1)u is initial surplus as n=0, and the insurer's premium income c is a constant at per unit time. means the ith claim size, and is the number of claim arriving in . which is the aggregate claims process in .Gerber[2, 3] and Shiu[4] had done lots of work to study this classical model in the end of the 1980's. Cheng Shixue, Wu Biao[9] and Zhu Rendong[11] also have taken a profound research on classical risk model in fully discrete setting in China and gotten recursive solutions, transform solutions and explicit solutions of the probability of ultimate ruin, the probability laws of the surplus immediately before ruin and the deficit at ruin. In addition, they derived the Lundberg upper bound of the probability of ultimate ruin for arbitral initial surplus. Sun Lijuan, Gu Lan[10], Wang Liming, Jin Heng[27], Gong Richao and Li Fengjun[26, 28] have generalized this classical model, they have considered that the premium income is a Poisson process and given the formula and the inequality of ruin probability. Yang Shanchao[12] have derived the approximate calculation of under compound binomial model. Wu Rong, Du Yonghong[20], Sun Lijuan, Gu Lan[21] have discussed the classical model with constant interest force and gotten the ruin probability of this model, the series expansion and the integral equation of the probability of ultimate ruin and the surplus immediately before ruin. When the number of premium income and the premium are random variables, Zou Hui, Zhu Yonghua[17] proved the inequality of ruin probability under the condition of the individual claim, X and the premium received every time, Y both following exponential distribution. In addition, Sun Lijuan, Gu Lan[10, 13] have done lots of work on stochastic simulation of ruin probability. The main aim of this paper is to generalize compound binomial model and make it to be the double binomial model. Premium income become a random variable following a discrete distribution. , (1.2)in this model, the second term (n) of the classical model(1.1) is generalized as , which means the aggregate number of policies in arrives the insurance company. For the proof, the main assumption are:The individual claim sizes are positive independent and identically distributed random variables with common distribution function F(x) and(2) is the claim number process for , ;(3) is the aggregate policies process, for , (4), are independent binomial process, and independent of ;(5) The condition a insurance company can manage in gear is .Now define the time of ruin as and the ruin probability as The Main ResultsTheorem 3.2 For , then andwhere R is the adjustment coefficient, and satisfies .Theorem 4.1 On the basic assumption of the model, for the ruin probability will satisfy where Theorem 5.1 Let , for , then .Theorem 5.2 Let , then , as , Theorem 6.2 Letthen.Theorem 7.1 As claim size is positive and integer random variable, let ;andfor;, then the ruin probability of the insurance company will be:where For the generalization of this model, this paper tries to further extend the model, assuming the premium is not constant when policies are arriving, but independent identically distributed random variable series , with common distribution function , and, , are indepe...