Collaborative Research: Applied Probability and Time Series Modeling

An investigation of the properties of Levy-driven CARMA (continuous-time ARMA) processes will be undertaken and efficient methods of inference developed. The results will be applied to the study of stochastic volatility models with Levy-driven CARMA volatility and to the further study of COGARCH models. Time series in which the parameters are constant over time-intervals between ``change-points'' constitute an important class of non-stationary time series which has been found particularly useful in hydrology, seismology and finance. Properties and applications of a new estimation technique based on the minimization of the minimum description length of a model that includes the number of change-points and their locations as parameters will be developed and extended to cover a general class of processes with structural breaks of various types. Estimation techniques for all-pass models driven by non-Gaussian noise will also be developed. These techniques, including maximum likelihood and minimum dispersion estimation, will be applied to the problem of identification and estimation for non-causal or non-invertible ARMA models.Adaptive techniques for efficient estimation of such models will be explored. In the last fifteen years, there has been a widely-recognized need for the development of new models and techniques for the analysis of time series data from scientific, engineering, biomedical, and financial applications. Some of the features required of these new models are nonlinearity, complex dependence structures, strong deviations from normality and non-stationarity. The current proposal addresses these needs. It seeks to enhance understanding of the physical and economic processes represented by the models. The development of efficient estimation and simulation techniques will be an essential component of the research.