Strong consistency of nonparametric kernel estimators for integrated diffusion process

Abstract

The asymptotic properties of nonparametric kernel estimators of diffusion process and integrated diffusion process were studied by scholars through using the theories of local time, giving the properties of consistency and asymptotic normality for nonparametric kernel estimators under appropriate conditions, but not property of strong consistency for integrated diffusion process. Instead of using the local time method, the paper applies the moment inequality of the ρ-mixing sequence to prove the strong consistency of the nonparametric kernel estimators in the integrated diffusion process. Our theorem conditions are mild and canonical, and some of them improve on the existing corresponding conditions. In numerical simulations and analysis of data from real applications, the nonparametric kernel estimators can capture well the variation characteristics of drift coefficient and diffusion coefficient, and that it is possible to fit parametric models with such characteristics, so that the economic interpretation of the models can be obtained.

Keywords:

• Strong consistency
• nonparametric kernel estimators
• integrated diffusion process

MATHEMATICS SUBJECT CLASSIFICATION (2000):

• 60J60
• 62G20

Authors’ contributions

All authors carried out the mathematical studies. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Additional information

Funding

This research was supported by the Guangxi Natural Science Foundation (No. 2022GXNSFAA035516), the Natural Science Foundation of China (No: 11461009) and the Important Natural Science Foundation of Colleges and Universities of Anhui Province (No. KJ2020A0122).