分数CIR过程统计行为的数值模拟  

Numerical Simulation of Statistical Behavior for Fractional Cox-Ingersoll-Ross Process

作　　者：刘博文 张静 陈晓鹏 LIU Bowen;ZHANG Jing;CHEN Xiaopeng(Business School,Nanfang College Guangzhou,Guangzhou,510000;Department of Mathematics,Shantou University,Shantou,515063)

机构地区：[1]广州南方学院商学院,广州510000 [2]汕头大学数学系,汕头515063

出　　处：《应用概率统计》2024年第1期1-17,共17页Chinese Journal of Applied Probability and Statistics

基　　金：广东省普通高校特色创新项目(批准号:2022KTSCX037);国家自然科学基金项目(批准号:12271185、11501344)资助。

摘　　要：Cox-Ingersoll-Ross (CIR)过程在金融领域有着重要的应用,本文主要对分数CIR过程的统计行为进行了模拟和探讨.由于该过程不存在解析解,拟采用两种不同随机函数wfbm和fbmld来模拟分数布朗运动,利用Euler-Maruyama (EM)方法模拟分数CIR过程的期望和方差.由于分数CIR过程的分布不能用福克–普朗克方程(Fokker-Planck equation)的解来表示,文章模拟了分数CIR过程的经验分布,并得到了随时间变化时的经验分布变化情况.为了进一步验证该Euler-Maruyama(EM)算法和比较两种随机函数的优越性,模拟了可以转化为向后欧拉法的分数Cox-Ingersoll-Ross(CIR)模型和具有解析解的分数Ornstein-Uhlenbeck (OU)模型,通过比较图像和数据,发现采用函数fbmld模拟的分数CIR过程的期望以及方差和理论上的期望以及方差具有较强的拟合程度.Cox-Ingersoll-Ross(CIR)process is an important tool to study stochastic interest rate and stochastic volatility infinancial market.The statistical behavior of fractional CIR process is mainly simulated and discussed in this paper.Since there is no analytical expression of the CIR process,two different functions wfbm and fbmld are used to simulate the fractional Brownian motion,and the Euler-Maruyama(EM)method is used to investigate the expectation and variance of the fractional CIR process.Because the distribution of fractional CIR process can not be expressed by the solution of Fokker-Planck equation,the empirical distribution of fractional CIR process is simulated,and the change of empirical distribution with time is obtained.In order to further verify the algorithm and compare the advantages of the two different algorithms,a backward Euler type scheme of the CIR model and the fractional Ornstein-Uhlenbeck(OU)model with analytical solution is carried out.By comparingfigure and table,it is found that simulation by the function fbmld have a very highfitting precision with the theoretical analytical solution with expectation and variance.

关 键 词：CIR过程 OU过程 EM方法 分数布朗运动 随机微分方程 

分 类 号：O211.64[理学—概率论与数理统计]