Copulas and stochastic processes

The modelling of dependence relations between random variables is one of the most widely studied subjects in probability theory and statistics. The most prominent models, particularly in practice, are presumably those which are based on correlation structures using second order moments. However, it is well known that only linear dependence can be captured this way and that it is characterizing only for special classes of distributions, for example the normal distribution. One possibility to avoid these limitations is to use copulas. These are multivariate distribution functions on the unit square with marginals being uniformly distributed on the unit interval. They separate the dependence structure from the influence of the marginals. Although most of the current research in copulas is done for spatial dependence (in the sense of a number of variables at the same time), copulas equally offer the possibility to analyze the intertemporal dependencies of stochastic processes. This was first observed by W.F. Darsow, B. Nguyen und E.T. Olsen (1992), who connected copulas to Markov processes. This thesis extends the relationship between copulas and stochastic processes into new directions and shows that the copula approach can provide new structural insights into stochastic processes. Known properties such as strict stationarity, symmetry or equivalence of processes are characterized. Sufficient conditions such that the process is stochastically continuous are derived by copulas and provide a new structural understanding of this type of continuity. Additionally, it is shown by example of EFGM processes and records that copulas may shorten existing proofs or make them more elementary. The technique of time transformation enables us to derive the copula structure of continuous local martingales which can then be used to generate new processes with a given intertemporal dependence structure.

Grantor dc:publisher

Publikationsserver der RWTH Aachen University

Year dc:date

2003

Author dc:creator

• Schmitz, Volker

Contributors dc:contributor

• Bock, Hans-Hermann

• info:eu-repo/classification/ddc/510
• Kopula <Mathematik>
• Stochastischer Prozess
• Mathematik
• Copulas
• Stochastic Processes
• Dependence Structures

Statement dc:rights

• info:eu-repo/semantics/openAccess

Language dc:language

eng

Identifier

https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121057%22

OAI identifier oai:identifier

oai:publications.rwth-aachen.de:59254