Two methods of estimation of the drift parameters of the Cox-Ingersoll-Ross process: continuous observations

We consider a stochastic differential equation of the form $dr_t = (a - b r_t) dt + \sigma\sqrt{r_t}dW_t$,where$a$,$b$and$\sigma$ 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 $\{r_t,t\in [0,T]\}$. 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 $a$, $b$, $\sigma$. The quality of the estimators is illustrated by simulation results.