Asymptotic properties for the parameter estimation in stochastic (functional) differential equations with Hölder drift

Abstract

In this paper, by taking Zvonkin's transformation, we investigate parameter estimation for a class of multidimensional stochastic differential equations with small perturbation parameters in diffusion coefficients, where the drift coefficients not only have an unknown parameter θ but also are Hölder continuous. These processes may enhance the applicability of our results to considerable practical models. Due to the irregular drift, the primary challenge is dealing with the mean square error between an accurate and numerical solution. Under these settings, we demonstrate the consistency and asymptotic normality of error concerning the least squares estimator in probability when stepsize $\delta \to 0$ and small parameter $\epsilon \to 0$ simultaneously. Moreover, we extend the results to the case of stochastic functional differential equations.

Keywords:

• Zvonkin's transformation
• Hölder drift
• least squares estimation
• consistency
• asymptotic normality

AMS Subject Classifications:

• 62F12
• 62M05
• 60H30
• 60J70

Acknowledgments

At this place we would like to thank our referees. Their detailed remarks and suggestions have greatly improved the content and presentation of our paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of Y. Hu would like to thank the financial support from the China Scholarship Council (CSC) and the BIT Research and Innovation Promoting Project (Grant No. 2022YCXZ037). The research of F. Xi was supported by the National Natural Science Foundation of China (Grant No. 12071031). The research of M. Zhu was supported by the National Natural Science Foundation of China (Grant No. 11901188).