Alexander Schnurr

Generalization of the Blumenthal-Getoor index to the class of homogeneous diffusions with jumps and some applications

We introduce the probabilistic symbol for the class of homogeneous diffusions with jumps (in the sense of Jacod/Shiryaev). This concept generalizes the well-known characteristic exponent of a Levy process. Using the symbol, we introduce eight indices which generalize the Blumenthal- Getoor indexand the Pruitt index �. These indices are used afterwards to obtain growth and Holder conditions of the process. In the future, the technical main results will be used to derive further fine properties. Since virtually all examples of homogeneous diffusions in the literature are Markovian, we construct a process which does not have this property.

The Euler Scheme for Feller Processes

We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented. In particular, the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of Lévy processes and, thus, the Euler scheme can be used for simulation by applying standard techniques from Lévy processes.

A Criterion for Invariant Measures of Ito Processes based on the Symbol

An integral criterion for the existence of an invariant measure of an Ito process is developed. This new criterion is based on the probabilistic symbol of the Ito process. In contrast to the standard integral criterion for invariant measures of Markov processes based on the generator, no test functions and hence no information on the domain of the generator is needed.

Testing for Structural Breaks via Ordinal Pattern Dependence

ABSTRACT We propose new concepts to analyze and model the dependence structure between two time series. Our methods rely exclusively on the order structure of the data points. Hence, the methods are stable under monotone transformations of the time series and robust against small perturbations or measurement errors. Ordinal pattern dependence can be characterized by four parameters. We propose estimators for these parameters, and we calculate their asymptotic distributions. Furthermore, we derive a test for structural breaks within the dependence structure. All results are supplemented by simulation studies and empirical examples. For three consecutive data points attaining different values, there are six possibilities how their values can be ordered. These possibilities are called ordinal patterns. Our first idea is simply to count the number of coincidences of patterns in both time series and to compare this with the expected number in the case of independence. If we detect a lot of coincident patterns, it would indicate that the up-and-down behavior is similar. Hence, our concept can be seen as a way to measure nonlinear “correlation.” We show in the last section how to generalize the concept to capture various other kinds of dependence.

Comparison of time-inhomogeneous Markov processes

Abstract Comparison results are given for time-inhomogeneous Markov processes with respect to function classes with induced stochastic orderings. The main result states the comparison of two processes, provided that the comparability of their infinitesimal generators as well as an invariance property of one process is assumed. The corresponding proof is based on a representation result for the solutions of inhomogeneous evolution problems in Banach spaces, which extends previously known results from the literature. Based on this representation, an ordering result for Markov processes induced by bounded and unbounded function classes is established. We give various applications to time-inhomogeneous diffusions, to processes with independent increments, and to Lévy-driven diffusion processes.