Collaborative Research: Applied Probability and Time Series Modeling

将对Levy驱动的CARMA(连续时间ARMA)过程的性质进行研究，并开发有效的推理方法。这些结果将被应用于研究具有Levy驱动的CARMA波动率的随机波动率模型，这些模型的应用超越了金融、湍流和一些神经科学过程。非平稳时间序列是一类重要的非平稳时间序列，在水文学、地震学、神经科学、环境科学和金融等领域具有重要的应用价值。一种基于最小化模型最小描述长度的新估计技术的性质和应用将被开发和推广，以涵盖具有结构突变的一般类型的过程，该模型包括变点的数目及其位置作为参数。希望这项技术也能适用于检测加性和创新离群值。多变量时间序列的线性和非线性模型，以期对时间脑动力学进行建模，也将在这项研究提案中发挥重要作用。这些模型包括可能的非线性向量自回归和一类不一定是因果的向量自回归。后一类虽然是线性的，但表现出以前只与非线性模型有关的特征，并允许在未来冲击的一个或多个组成部分的相关性意义上具有前瞻性的可能性。在过去的15年里，人们普遍认为需要开发新的模型和技术来分析来自科学、工程、生物医学、金融和神经科学应用的时间序列数据。这些新模型所要求的一些特征是非线性、复杂的相依结构、强偏离正态分布和非平稳性。在神经科学、环境和金融建模中，也存在对包含这些特征的连续时间模型的需求。目前的提案满足了这些需求。它寻求加强对模型所代表的物理、生物医学和经济过程的理解。开发有效的估计和模拟技术将是这项研究的重要组成部分。

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.