Change-Point Detection for Discretely Sampled Diffusion Processes

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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.