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 better understand and model the real-time market dynamics, to better assess market quality, and to better regulate financial markets.
