This project considers parameter estimation and model testing for a wide range of continuous-time processes used in stochastic modeling. They include processes which have discontinuous sample paths, are multivariate, only partially observed and are subject to long range dependence. Despite being wide ranging, a common opportunity offered by these processes is that the conditional characteristic functions are available for commonly used processes in stochastic modeling. The investigators will develop a framework of statistical inference for stochastic processes based on the conditional characteristic functions. They will consider both parameter estimation and model testing by employing three modern nonparametric tools of statistical inference: the kernel smoothing method, the empirical likelihood and the bootstrap resampling method. Developing efficient parameter estimators and robust test procedures for stochastic processes are the goals of this project.
Continuous time stochastic processes defined by stochastic differential equations have long been used to model dynamic stochastic systems arising in physics, biology and other natural sciences. One latest surge of interest on these processes comes from molecular biology in modeling the dynamics of proteins as part of an effort to understand how energy transfer and conversion happen within a biological cell. Perhaps the most eminent use of the continuous time stochastic processes in the last two decades has been in finance following the works of Merton (1971) and Black and Scholes (1973) which established the foundation of option pricing theory for modern finance. Analysts and practitioners of these continuous-time stochastic models are increasingly aware that no matter how rich and powerful these models are for modeling a stochastic system, their applicability will be largely limited if model parameters cannot be estimated and the validity of the models cannot be confirmed with empirical data. The intellectual merit of the project is in establishing a statistical inference framework for stochastic processes commonly used in stochastic modeling these days. The broader impacts of the proposed research are (1) being on the cutting edge of statistics, probability and stochastic system modeling, (2) producing opportunities for collaboration and partnerships with practitioners from a range of disciplines, and (3) extending the educational experience of students with an opportunity to be deeply involved with stochastic modeling and applications.
