Title: Local asymptotic properties for Cox-Ingersoll-Ross process with discrete observations,
Abstract: In this paper, we consider a one-dimensional Cox-Ingersoll-Ross (CIR) process
whose drift coefficient depends on unknown parameters. Considering the process
discretely observed at high frequency, we prove the local asymptotic normality
property in the subcritical case, the local asymptotic quadraticity in the
critical case, and the local asymptotic mixed normality property in the
supercritical case. To obtain these results, we use the Malliavin calculus
techniques developed recently for CIR process together with the $L^p$-norm
estimation for positive and negative moments of the CIR process. In this study,
we require the same conditions of high frequency $\Delta_n\rightarrow 0$ and
infinite horizon $n\Delta_n\rightarrow\infty$ as in the case of ergodic
diffusions with globally Lipschitz coefficients studied earlier by Gobet
\cite{G02}. However, in the non-ergodic cases, additional assumptions on the
decreasing rate of $\Delta_n$ are required due to the fact that the square root
diffusion coefficient of the CIR process is not regular enough. Indeed, we
assume $\frac{n\Delta_n^{\frac{3}{2}}}{\log(n\Delta_n)}\to 0$ for the critical
case and $n\Delta_n^2\to 0$ for the supercritical case.