Asymptotic expansion, statistical inference and their applications

构造了具有跳跃的随机微分方程的拟似然分析。提出了连续扩散型过程的自适应方法。这为实现提供了基础。研究了随机回归模型的变点问题，并通过混合型高斯泛函给出了其极限定理。给出了非同步协方差估计量（Hayashi-Yoshida估计量）渐近混合正态性的证明。构造了一个超前-滞后问题，并证明了超前-滞后估计器对于一个带滞后的半鞅模型的收敛速率。提出了具有混合正规极限的鞅的渐近展开式理论，并将其应用于已实现的波动。

A quasi likelihood analysis(QLA) was constructed for stochastic differential equations with jumps. Adaptive methods for continuous diffusion type processes were proposed. This gives a basis for implementation. The change point problem was studied for a stochastic regression model, and the limit theorem was presented through a mixture type Gaussian functional. Proof of the asymptotic mixed normality of the nonsynchronous covariance estimator(Hayashi-Yoshida estimator) was published. A lead-lag problem was framed and we proved a convergence rate of our lead-lag estimator for a semimartingale model with lag. A theory of asymptotic expansion for a martingale with mixed normal limit was initiated and applied to the realized volatility.