Non parametric estimation of transition density for second-order diffusion processes

Abstract

The transition density of the diffusion process plays an important role in calculating the dynamic characteristics of the underlying variables as well as the model estimation. In this article, we combine the idea of conditional probability density function with non parametric kernel regression, and introduce the kernel estimation of joint density function and marginal density function, then construct the non parametric kernel estimator of the transition density of second-order diffusion process based on discrete observational samples. In order to obtain the asymptotic properties of the new kernel estimator, we analyze the asymptotic expectation and asymptotic variance of the proposed estimator under some mild conditions. Finally, the consistency and asymptotic normality of the new proposed non parametric estimator of the transition density function are proved.

KEYWORDS:

• Second-order diffusion process
• Kernel estimation
• transition density
• weak convergence

Acknowledgments

The authors would like to thank the referees for their valuable suggestions, which greatly improved the structure and the presentation of the paper.

Additional information

Funding

This work was supported by the National Nature Science Foundation of China under Grant numbers 11461032 and 11401267, and Jiangxi University of Science and Technology under the Program of Qingjiang Excellent Young Talents.