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

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.

Second-order linear ordinary differential equation with non-local integral term inside are major discussed in this thesis. Firstly, we canvass the existence and uniqueness of the solution of the second-order ordinary integro-differential equation. By variable substitution and exchanging sequence of repeated integral, the non-local integral term can be simplified, the equation can be transformed into system of equations. Then, the proof of existence and uniqueness of the solution of equations are completed here. Second, we analyze the structure of the solution, also we give the solution form of the equation. At last, we point out the application of the ordinary integro-differential equation through deduction. It can be used in ruin probabilities＇ of an insurance company. We focus our energies upon the application of the second-order equation. Furthermore, the explicit expressions for the integro-differential equations will be presented when the claims are exponentially distributed.