Asymptotically efficient estimation for diffusion processes with nonsynchronous observations

TL;DR

This paper develops an asymptotically efficient maximum-likelihood-type estimator for diffusion processes with nonsynchronous high-frequency observations, relevant for financial data analysis, and proves its consistency, normality, and efficiency.

Contribution

It introduces a new estimator for diffusion processes with nonsynchronous data and establishes its asymptotic properties, including efficiency, under general sampling conditions.

Findings

Estimator is consistent and asymptotically normal.

Estimator achieves asymptotic efficiency.

Results apply to high-frequency financial data with mixing sampling schemes.

Abstract

We study maximum-likelihood-type estimation for diffusion processes when the coefficients are nonrandom and observation occurs in nonsynchronous manner. The problem of nonsynchronous observations is important when we consider the analysis of high-frequency data in a financial market. Constructing a quasi-likelihood function to define the estimator, we adaptively estimate the parameter for the diffusion part and the drift part. We consider the asymptotic theory when the terminal time point $T_n$ and the observation frequency goes to infinity, and show the consistency and the asymptotic normality of the estimator. Moreover, we show local asymptotic normality for the statistical model, and asymptotic efficiency of the estimator as a consequence. To show the asymptotic properties of the maximum-likelihood-type estimator, we need to control the asymptotic behaviors of some functionals of the sampling scheme. Though it is difficult to directly control those in general, we study tractable sufficient conditions when the sampling scheme is generated by mixing processes.