Abstract

We consider a stochastic differential equation of the form $d{r}_t=(a-b{r}_t)dt+\sigma \sqrt{r_t}d{W}_t,$ where a, b and σ are positive constants. The solution corresponds to the Cox–Ingersoll–Ross process. We study the estimation of an unknown drift parameter (a, b) by continuous observations of a sample path $\left\{r_t,t\in [0,T]\right\}.$ First, we prove the strong consistency of the maximum likelihood estimator. Since this estimator is well-defined only in the case $2a>{\sigma }^2,$ we propose another estimator that is defined and strongly consistent for all positive a, b, σ. The quality of the estimators is illustrated by simulation results.

中文翻译：

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Cox-Ingersoll-Ross 过程漂移参数的两种估计方法：连续观测

摘要

我们考虑以下形式的随机微分方程$d{r}_{吨}=(\mathrm{一个}-b{r}_{吨})d吨+\sigma \sqrt{r_{吨}}d{W}_{吨},$其中a、b和σ是正常数。该解对应于 Cox-Ingersoll-Ross 过程。我们通过对样本路径的连续观察来研究未知漂移参数 ( a , b )的估计$\left\{r_{吨},吨\in [0,吨]\right\}.$首先，我们证明了最大似然估计量的强一致性。因为这个估计量仅在这种情况下是明确定义的$2\mathrm{一个}>{\sigma }^2,$我们提出了另一个对所有正的 a、b、σ都有定义且高度一致的估计量。仿真结果说明了估计器的质量。