Strong consistency of parameter estimation for the CIR integrated diffusion process with long-span high-frequency data

In this article, we utilize the contrast function of the integrated diffusion process to derive parameter contrast estimators for the Cox-Ingersoll-Ross integrated diffusion process and discuss the asymptotic properties of the estimators using the mixing properties of the process. We first give the tail probability exponential inequality of the mixing long-span high-frequency data. Then, applying this inequality, we prove the strong consistency of the contrast estimators for the drift parameters and diffusion parameter of the CIR integrated diffusion process. In order to obtain these asymptotic properties, we require relatively low order r of the negative power moment E|Xt−1|r of the diffusion process. In the existing literature, r > 9 is required in order to obtain weak consistency of the estimators. However, in our theorems, only r > 2 is required for the strong consistency of the diffusion parameter estimator, while r > 4 for the strong consistency of the drift parameter estimators. In the simulation analysis, we use the Euler discretization algorithm and the Milstein discretization algorithm, and the results show that the optimal sampling interval is around n−1/3. We use the daily log price data of the CSI 300 Index to estimate the CIR integrated diffusion model and construct a Bootstrap prediction model for the log price. The results show that the model has a good predictive performance.