带非局部积分项常微分方程的讨论及其应用Discussion and application of ordinary differential equation with non-local integral term
朱一峰;边保军;
摘要(Abstract)：
研究带非局部积分项的二阶线性常微分方程及其在金融保险上的应用.首先讨论带非局部积分项的二阶常微分方程解的存在唯一性,通过变量代换和累次积分交换积分顺序将非局部项简化,将方程化为方程组,然后完成了对方程组解的存在唯一性的证明.接着分析了带非局部项的二阶常微分方程解的结构,给出了方程解的形式.最后通过推导,指出带非局部项的线性常微分方程在保险公司的破产概率研究中的应用,重点放在二阶方程的应用上,并且在某一特定情况下,举出了一个可以给出解析解的例子.
关键词(KeyWords)： 非局部积分项;二阶常微分方程;破产概率
基金项目(Foundation):
作者(Authors): 朱一峰;边保军;
参考文献(References)：
- [1]Gerber H U.An extension of the renewal equation and its application in the collective theory of risk[J].Scandinavian Actuarial Journal,1970(3):205-210.
- [2]Albrecher H,Borst S,Boxma O,et al.The tax identity in risk theory-a simple proof and an extension[J].Insurance:Mathematics and Economics,2009,44:304-306.
- [3]Wang W,Ming R,Hu Y.On the expected discounted penalty function for risk process with tax[J].Statist.Probab.Lett.,2011,81:489-501.
- [4]王高雄,周之铭,朱思铭,等.常微分方程[M].2版.北京:高等教育出版社,1983.
- [5]Dufresne F,Gerber H U.Risk theory for the compound Poisson process that is perturbed by diffusion[J].Insurance:Mathematics and Economics,1991,10:51-59.
- [6]Revuz D,Ypr M.Continuous Martingales and Brownian Motion[M].Berlin:Springer,1991.
- [7]Port S,Stone C.Brownian Motion and Classical Potential Theory[M].New York:Academic Press,1978.
- [8]Wang G,Wu R.Some distributions for classical risk process that is perturbed by diffusion[J].Insurance:Mathematics and Economics,2000,26:15-24.
- [9]Gerber H U,Shiu E.From Ruin Theory to Option Pricing[M].Cairns:AFIR Colloquium,1997.