Change-Point Detection for Discretely Sampled Diffusion Processes

虽然在离散时间随机变量序列的情况下，例如回归模型，已经存在许多用于各种模型变化问题（变点问题）的统计方法，但是在随机微分方程（扩散）的（连续时间，非确定性）解中，这种变点的统计检测还很少受到关注。通常，变化点表示观察期中确定模型的参数发生变化的时间点。这个项目的目的是提供一个贡献，以弥合离散时间和连续时间的方法之间的差距差距检测统计模型中的变点。为此，它旨在扩展自己的博士开发的统计检验程序。论文以设置适合应用的问题为目标。例如，在一个示例中，随机微分方程代表了用于金融数据或生理过程（例如葡萄糖-胰岛素平衡和神经过程的动力学）的数学建模的现代工具。在生理模型方面，人们可以从哥本哈根的大量经验中受益，以便使数学假设适应应用领域的要求。重点是获得必要的信息，从离散的，即多个，数据的扩散的连续时间演化。此外，以下概括的模型计划：多维参数扩散也多维参数和扩散只存在于整个空间的子集。在本项目的背景下，预期的结果是几个强大的测试程序，包括一个监测程序（顺序测试），允许在观察期间已经评估的参数。

While in the case of discrete-time series of random variables, e.g. regression models, a lot of statistical procedures for all kinds of problems of a model change (change-point problem) already exist, it has been payed little attention to the statistical detection of such change-points in (continuous-time, non-deterministic) solutions of stochastic differential equations (diffusions) yet. Typically, a change-point denotes the time point in the observation period where the parameters determining the model change. Aim of this project is to provide a contribution to bridging the gap between discrete-time and continuous-time approaches for detecting change-points in statistical models. Toward this end, it is intended to extend the developed statistical test procedures of the own Ph.D. thesis to settings of the problem suitable for applications. E.g., stochastic differential equations represent a modern tool for mathematical modelling of financial data or of physiological processes such as the dynamics of the glucose-insulin balance and neuronal processes. Primarily at physiological models, one can benefit from the considerable experience in Copenhagen in order to adapt the mathematical assumptions to the requirements in fields of application. The focus is on obtaining the necessary information about the continuous-time evolution of the diffusion from discrete, i.e. finitely many, data. Moreover, the following generalisations of the model are planned: multidimensional parametric diffusions with also multidimensional parameters and diffusions which only exist in subsets of the whole space. In the context of this project, the intended results are several powerful test procedures including a monitoring procedure (sequential test) allowing an evaluation of the parameters already during the observation period.