Ait-Sahalia利率模型数值解收敛性分析

Convergence analysis of numerical solution of Ait-Sahalia interest rate model MIN Yingxiao,JI Yanting,WANG Yingying,FANG Qiquang

【目的】为研究一类高度非线性的广义Ait-Sahalia利率模型,对其数值解的收敛性进行证明。【方法】首先引入迭代方法证明方程存在唯一的全局正解;然后从经典欧拉(Euler-Maruyama,EM)数值格式出发,得到了广义Ait-Sahalia利率模型的驯服(tamed)欧拉数值解;最后修正方程系数所满足的条件,证明方程的驯服欧拉数值解依概率收敛于方程的解析解。【结果】对于漂移项和扩散项都高度非线性的随机微分方程,通过改进经典欧拉方法及处理方程漂移项和扩散项的系数条件,可获得具有依概率收敛性质的数值解。【结论】本研究结果可推广至其他类型的利率模型数值解研究,对金融衍生品分析和定价具有一定的指导意义。

[Objective] A generalized Ait-Sahalia interest rate model, being highly nonlinear, was investigated to demonstrate the convergence of its numerical solution. [Method] Firstly, an iterative method was introduced to prove the existence of a unique global positive solution to the equation;then, the tamed Euler-Maruyama(EM)numerical solution of the generalized Ait-Sahalia interest rate model was derived from the classical EM numerical format;finally, the conditions satisfied by the coefficients of the equation were modified to prove that the tamed EM numerical solution of the equation converges with probability to the analytical solution of the equation. [Result] For the stochastic differential equation with highly nonlinear drift and diffusion terms, the numerical solution with probabilistic convergence can be obtained by improving the classical EM method and dealing with the coefficient conditions of the drift and diffusion terms of the equation. [Conclusion] The research results can be extended to the numerical solution research of other types of interest rate models, which is of guiding significance for the analysis and pricing of financial derivatives.