Change-Point Detection for Discretely Sampled Diffusion Processes
Final Report Abstract
The project “Change-Point Detection for Discretely Sampled Diffusion Processes” was concerned with the design of several statistical procedures for testing time-series on structural breaks (“change-points”) and especially for monitoring their structural stability online. The investigated time-series are multidimensional stochastic processes depending on also multidimensional parameters. We say that a process suffers from a structural break if there exists a point in time, the change-point, at which a parameter value suddenly changes. This break implies a change of the distribution of the process. To put it in another way, the question in change-point analysis is whether the observed fluctuations of a stochastic process always follow the same distribution or whether there is a structural change behind it. It turned out that it was possible to prove deeper and much more general results than expected. That is, the designed procedures are applicable to a considerably larger class of processes than only to diffusion processes. This class covers discrete-time as well as continuous-time models. Examples for discrete-time processes treatable by the developed methodology are autoregressive time-series. Examples for continuoustime models are stochastic differential equations (SDEs) and even complex transformations of SDEs like integrated diffusions used in climatology to model ice-core data sets. Bridging the gap between dicrete and continuous approaches in change-point analysis was one of the most important goals of the project. Having applications in finance in mind, simulation studies with a version of the Black-Scholes SDE (also called “GARCH diffusion model”) were successfully performed for all proposed procedures. The GARCH diffusion is an advanced way to model stochastic volatilities of financial assets. In addition to the parallel applicability to autoregressive time-series and SDEs, the promising simulation results give rise to hope on a diversity of potential applications in financial mathematics.
