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.
翻译:当系数是非随机的并且观测以非同步的方式发生时,我们研究扩散过程的最大可能性类型的估计。当我们考虑在金融市场分析高频数据时,非同步的观测问题是十分重要的。构建一个准相似功能来定义估计器,我们适应地估计扩散部分和漂移部分的参数。当终点点$T_n$和观测频率达到无限时,我们考虑无症状理论,并显示估计器的连贯性和无同步性正常性。此外,我们显示统计模型的局部性非同步性正常性,以及作为结果的估测器的无症状效率。要显示最大可能性类型估测器的无症状特性,我们需要控制取样方法中某些功能的无症状行为。虽然很难直接控制这些功能,但我们在混合过程产生采样计划时研究足够的可移动性条件。