标题： 通过CIR框架对CKLS模型中的参数估计的精确离散化方法 显示英文标题

标题： An Accurate Discretized Approach to Parameter Estimation in the CKLS Model via the CIR Framework

摘要： 本文提供了对利率模型中参数估计和渐近行为的见解，主要关注Cox-Ingersoll-Ross (CIR)过程及其扩展——更一般的Chan-Karolyi-Longstaff-Sanders (CKLS)框架 ($\alpha\in[0.5,1]$)。CIR过程被广泛用于建模具有均值回归特性的利率。CIR模型的扩展，CKLS模型作为分析更复杂动态的基础案例。我们采用Euler-Maruyama离散化方法将这些模型的连续时间随机微分方程(SDEs)转换为便于使用线性回归技术进行高效模拟和参数估计的离散形式。我们建立了漂移和波动率参数估计量的强一致性和渐近正态性，为参数估计过程提供了理论基础。此外，我们通过检查涉及多项式漂移和扩散项的广义SDE相关尺度和速度密度函数，探讨了这些模型在零和无穷远处不可达性背景下的边界行为。此外，我们推导了CKLS框架内平稳分布存在的充分条件及相应的平稳密度函数；并讨论其对模型参数的依赖性对于$\alpha\in[0.5,1]$。 显示英文摘要

摘要： This paper provides insight into the estimation and asymptotic behavior of parameters in interest rate models, focusing primarily on the Cox-Ingersoll-Ross (CIR) process and its extension -- the more general Chan-Karolyi-Longstaff-Sanders (CKLS) framework ($\alpha\in[0.5,1]$). The CIR process is widely used in modeling interest rates which possess the mean reverting feature. An Extension of CIR model, CKLS model serves as a foundational case for analyzing more complex dynamics. We employ Euler-Maruyama discretization to transform the continuous-time stochastic differential equations (SDEs) of these models into a discretized form that facilitates efficient simulation and estimation of parameters using linear regression techniques. We established the strong consistency and asymptotic normality of the estimators for the drift and volatility parameters, providing a theoretical underpinning for the parameter estimation process. Additionally, we explore the boundary behavior of these models, particularly in the context of unattainability at zero and infinity, by examining the scale and speed density functions associated with generalized SDEs involving polynomial drift and diffusion terms. Furthermore, we derive sufficient conditions for the existence of a stationary distribution within the CKLS framework and the corresponding stationary density function; and discuss its dependence on model parameters for $\alpha\in[0.5,1]$.