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 that have applications that go beyond finance to turbulence and some neuroscience processes. 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, neuroscience, environmental science 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. It is hoped that this technique can also be adapted for detection of both additive and innovational outliers. Linear and nonlinear models for multivariate time series, with a view towards modeling temporal brain dynamics, will also play a major role in this research proposal. These models include a mixture of possibly nonlinear vector autoregressions and a class of not necessarily causal vector autoregressions. The latter class, although linear, exhibits features previously only associated with nonlinear models and allows for the possibility of foresight in the sense of dependence of one or more components of future shocks.
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, financial, and neuroscience applications. Some of the features required of these new models are nonlinearity, complex dependence structures, strong deviations from normality and non-stationarity. In neuroscience, environmental and financial modeling there is also a demand for continuous-time models which incorporate these features. The current proposal addresses these needs. It seeks to enhance understanding of the physical, biomedical, and economic processes represented by the models. The development of efficient estimation and simulation techniques will be an essential component of the research.