A0555
Title: Estimation of a pure-jump stable Cox-Ingersoll-Ross process
Authors: Elise Bayraktar - Universite Gustave Eiffel (France) [presenting]
Emmanuelle Clement - Universite Gustave Eiffel (France)
Abstract: A stable Cox-Ingersoll-Ross ($\alpha$-stable CIR) process defined by $dX_t=(a - b X_t)dt + \sigma X_{t}^{1/2} dW_t + \delta X_{t-}^{1/ \alpha} dL^{\alpha}_t, \quad X_0=x_0 >0,$ where $(L^{\alpha}_t)$ is a stable Levy process with non-negative jumps and jump activity index $\alpha \in (1,2)$ and $(W_t)$ is a standard Brownian motion. The pure jump case $\sigma=0$ is considered. The aim is to study the joint estimation of drift, scaling and jump activity parameters $(a, b \delta, \alpha)$ from high-frequency observations of the process on a fixed time period. The existence of a joint estimator of $(a, b, \delta, \alpha)$ is proven based on an approximation of the likelihood function, which is consistent and asymptotically conditionally Gaussian. Moreover, the uniqueness of the drift estimators is established, assuming that $\delta$ and $\alpha$ are known or consistently estimated. Easy-to-implement preliminary estimators of all parameters are proposed, and those are improved by a one-step procedure. The conclusion is by proposing an estimation method for the general case $\sigma>0$ based on the method of moments.
