Collaborative Research: Applied Probability and Time Series Modeling

对levy驱动的CARMA（连续时间ARMA）过程的性质进行了调查，并开发了有效的推理方法。研究结果将应用于levy驱动的CARMA波动率的随机波动率模型的研究，该模型的应用范围不仅限于金融，还包括湍流和一些神经科学过程。在变化点之间的时间间隔内参数恒定的时间序列构成了一类重要的非平稳时间序列，它在水文学、地震学、神经科学、环境科学和金融中特别有用。基于模型的最小描述长度的最小化的新估计技术的属性和应用，包括变化点的数量及其作为参数的位置，将被开发和扩展到涵盖具有结构断裂的一般类型的过程。希望该技术也可以适用于附加和创新异常值的检测。多元时间序列的线性和非线性模型，着眼于模拟时间大脑动力学，也将在本研究计划中发挥重要作用。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的混合。后一类虽然是线性的，但显示出以前只与非线性模型相关的特征，并允许在依赖未来冲击的一个或多个组成部分的意义上预见的可能性。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科学、工程、生物医学、金融和神经科学等领域的时间序列数据。这些新模型所要求的一些特征是非线性、复杂的依赖结构、偏离正态性强和非平稳性。在神经科学、环境和金融建模中，也需要结合这些特征的连续时间模型。当前的提案解决了这些需求。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。开发有效的估算和模拟技术将是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.