Statistical Analysis of the Filtering Models with Marked Point Process Observations: Applications to Ultra-High Frequency Data

本文从超高频数据的新视角出发，以金融市场微观结构理论为有力的理论依据，借助随机微积分的强大工具，研究了两类具有标点过程观测值的滤波问题。第一行是马尔可夫信号，第二行是由分数布朗运动驱动的长记忆信号。对于每条线，研究者系统地开发统计分析及其应用。研究者建立推断的统计基础：似然、后验分布、似然比和贝叶斯因子。通常，它们都是无限维的，并且由随机微分方程（如滤波方程）表征。这样的方程推导和重点是通过过滤的两个模型发展贝叶斯推理。此外，研究者研究了两种方法来构建高性能的计算算法的在线实施的统计分析。一种方法是马尔可夫链近似方法，另一种是粒子滤波或序列蒙特卡罗方法。这些精心设计的算法的一致性的数学基础建立。在世界所有主要金融市场中，关于增加的市场变量的超高频（或逐交易）数据广泛存在，过滤模型的动机来自市场微观结构理论。随着过滤模型及其建立的统计理论和工具，研究者开发了大量的金融经济应用。这些帮助市场微观结构研究人员更好地理解和模拟实时市场动态，更好地评估市场质量，更好地监管金融市场。

With a new view of ultra-high frequency data, strong motivation from financial market microstructure theory, and the powerful machinery of stochastic calculus, this proposal studies two lines of the filtering problems with marked point process observations. Line One has Markov signals and Line Two has long-memory signals driven by fractional Brownian motion. For each line, the investigator systematically develops the statistical analysis and its applications. The investigator establishes the statistical foundations for inference: the likelihoods, the posterior distribution, the likelihood ratio and the Bayes factors. Typically, they are all infinite-dimensional and are characterized by stochastic differential equations such as filtering equations. Such equations are derived and the focus is on developing Bayesian inference via filtering for the two models. Moreover, the investigator studies two approaches for constructing high-performance computing algorithms for the on-line implementation of the statistical analysis. One approach is the Markov chain approximation method, and the other is particle filtering or sequential Monte Carlo method. The mathematical foundations for the consistency of these carefully-designed algorithms are established. Ultra-high frequency (or trade-by-trade) data are widely available on increased market variables in all major world financial markets and the filtering models are motivated from market microstructure theory. With the filtering models and their established statistical theory and tools, the investigator develops substantial financial economic applications. These help market microstructure researchers to betterunderstand and model the real-time market dynamics, to better assess market quality, and to better regulate financial markets.