Applied Probability and Time Series Modelling

9972015本研究涉及与离散和连续时间非线性模型的理论和应用以及整数值数据模型相关的时间序列分析问题。 它关注的是这种模式的发展，研究其属性和调查系统的技术，以适应他们的数据。 也被认为是估计模型参数的大样本属性和基于模型的预测的性能。 许多线性时间序列分析的标准工具（例如样本自协方差函数）在基本过程为非线性时具有实质上不同的属性，并且当它们与非线性模型结合使用时必须考虑到这一点。 非线性连续时间阿尔马模型已被发现在金融数据建模中是有用的。 有效的估计这些过程和非高斯和多元扩展这些被调查，以允许观察到的金融时间序列的重尾行为，并允许相关的金融时间序列之间的依赖关系的研究。 另一种有前途的金融时间序列分析方法是使用非高斯噪声驱动的全通线性滤波器作为使用非线性模型的可能替代方案。 整数值时间序列具有广泛的存在性，例如每周发生的事故数或某些疾病的确诊病例数。 至于上面考虑的模型，目标是开发一个系统的方法来建模和预测这样的data.In过去的十年里，一直在稳步增长的认识，非线性时间序列模型提供了更好的表示许多经验观察到的时间序列比经典的线性模型。 然而，许多非线性模型的属性还没有被理解，有必要开发新的模型，伴随着有效的模型拟合程序。 与此同时，人们对连续时间时间序列模型的兴趣激增，部分原因是随机微分方程模型非常成功地应用于金融问题，布莱克-斯科尔斯期权定价公式的推导及其推广就是例证。 本建议涉及离散和连续时间的非线性模型的开发，特别强调这些模型在金融数据表示和预测中的应用。 另一类在应用中经常出现的时间序列，其系统分析相对不发达，是那些观测值为整数值的时间序列。 对于这些系列也提出了一个系统的方法来建模和预测。

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.