Applied Probability and Time Series Modelling

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9972015This research deals with problems in time series analysis related to the theory and application of non-linear models in both discrete and continuous time and to models for integer-valued data. It is concerned with the development of such models, the study of their properties and the investigation of systematic techniques for fitting them to data. Also considered are the large-sample properties of the estimated model parameters and the performance of model-based forecasts. Many of the standard tools of linear time series analysis (e.g. the sample autocovariance function) have substantially different properties when the underlying process is nonlinear and this must be taken into account when they are used in conjunction with nonlinear models. Nonlinear continuous time ARMA models have been found useful in the modeling of financial data. Efficient estimation for these processes and for non-Gaussian and multivariate extensions of these are to be investigated to allow for the observed heavy-tailed behavior of financial time series and to permit the study of the dependence between related financial time series. An alternative promising approach to the analysis of financial time series is to use all-pass linear filters driven by non-Gaussian noise as a possible alternative to the use of nonlinear models. Integer-valued time series are of wide occurrence, for example the weekly numbers of accidents or diagnosed cases of some disease. As for the models considered above, the objective is to develop a systematic approach to the modelling and forecasting of such data.In the last ten years there has been a steadily increasing realization that non-linear time series models provide much better representations of many empirically observed time series than the classical linear models. Many of the properties of non-linear models are however not yet understood and there is a need for the development of new models accompanied by efficient model-fitting procedures. At the same time there has been a surge of interest in continuous-time time series models, partly as a result of the very successful application of stochastic differential equation models to problems in finance, exemplified by the derivation of the Black-Scholes option-pricing formula and its generalizations. This proposal is concerned with the development of non-linear models in both discrete and continuous time with particular emphasis on the application of these models to the representation and forecasting of financial data. Another class of time series which arises frequently in applications and for which systematic analysis is relatively undeveloped are those in which the observations are integer-valued. For these series also a systematic approach to model-building and forecasting is proposed.