Research

Due to the variety of non-deterministic processes of our empirical world it is often the case that the probability mechanism of an underlying stochastic process displays a structural break on unknown jump discontinuities.

If for example the process describes a temporal development of a random system, a jump discontinuity indicates in the simplest case a specific point in time in or after which a sudden or gradual change of the distribution of random observables occurs. The distribution of observables can also change after several points in time. In this case we talk about a (finite) multidimensional jump discontinuity.

If however the process describes a random system using locations instead of points in time, a jump discontinuity constitutes a domain. Observations made within this domain follow a different distribution than those made outside the domain. These domains are infinite dimensional discontinuities, as long as there is no or little information about their form. Apart from jump discontinuities in space and time, there are also jump discontinuities on the parameter scale of distribution. Here, we can observe in a simple situation independent and identically distributed real random variables whose joint density shows a discontinuity.

In this case, the problem is to make an estimate of the jump discontinuity based on the observation of the stochastic process. Depending on the given question, the estimated value provides a time interval of a defective production of an item, duration of an epidemic, reconstruction of a noisy image or the division of patients into high- and low-risk groups. The task is now to develop estimation methods in the respective model. These will be optimised after an efficiency test using mathematical criteria. This happens through a derivation of limit theorems. The implementation of algorithms as computer programs enables us to carry out comprehensive simulation studies.

The research results of the chair are documented in numerous publications.