Statistical asymptotic theory for stochastic processes

研究了渐近展开理论，发展了随机过程的统计推断.作为混合性质的推广，引入了部分混合的概念，建立了部分混合过程的渐近展开理论。应用这一理论推导了一个具有长记忆解释随机过程的随机回归模型的展开式。由于二阶项与非中心极限定理有关，因此展开式是不规范的.利用抽象的Malliavin算子将Watanabe理论推广到一般情形，并导出了条件展开式作为双Edgeworth展开式.作为应用，给出了一个滤波问题的算法.提出了一种基于非同步观测的两个扩散过程相关系数的估计。这个问题是由财务数据分析引起的。证明了估计量的相合性.提出了一类带跳随机微分方程参数的M-估计，并证明了其渐近性质（相合性、渐近正态性和渐近展开性）.我们推导了一个以Levy驱动的Ornstein-Uhlenbeck过程为波动项的随机波动模型的展开公式。这使我们能够理解股票收益率的聚集高斯性和非高斯性.我们发表或准备的文件渐近展开的ε-马尔可夫过程，扩展下退化，和预测区域。

Theory of asymptotic expansions was investigated to develop the statistical inference for stochastic processes.1. As a generalization of mixing property, the notion of the partial mixing was introduced, and the theory of asymptotic expansion for partial mixing processes was established. This theory was applied to derive expansions for a stochastic regression model with a long-memory explanatory stochastic process. The resulting expansion is not standard in that the second-order term is related with the non-central limit theorem.2. An extension of the Watanabe theory to a general setting with an abstract Malliavin operator was build, and also conditional expansion formulas were derived as the double Edgeworth expansion. As an application, an algorithm for a filtering problem was provided.3. We proposed an estimator of the correlation coefficient between two diffusion processes based on non-synchronous observations. This problem was motivated by financial data analysis. Consistency of the estimator was proved.4. M-estimators of parameters in a stochastic differential equation with jumps were proposed and asymptotic behaviors (consistency, asymptotic normality and asymptotic expansion) were proved.5. We derived an expansion formula for a stochastic volatility model with the Levy-driven Ornstein-Uhlenbeck process as the volatility term. It enables us to understand the aggregation Gaussianity and non-Gaussianity of the stock returns.6. We published or prepared paper on asymptotic expansion for epsilon-Markov processes, expansion under degeneracy, and prediction regions.