Stochastic calculus for fractional Lévy processes and related processes

• Applicant: Professor Dr. Christian Bender
• Subject Area: Mathematics
• Term: from 2010 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 192622538

Stochastic processes can be considered as models for time dependent systems which are subjectto random infuences. A statistical analysis of many real world phenomena in such diverse fields as finance, econometrics, hydrology, or internet traffic, reveals long memory effects. Arguably, the best-studied stochastic processes with long range dependence are fractional Brownian motions (with Hurst parameter H > 1=2). Fractional Lévy processes are generalizations of fractional Brownian motions which capture the memory effects in a similar fashion but provide more flexibility concerning the modeling of the distribution. The aim of the project is to establish a stochastic calculus for fractional Lévy processes and related processes which can be obtained by a convolution of a deterministic kernel with a Lévy process. We will focus on stochastic integrals in the Skorokhod sense, whose zero expectation property is important for its interpretation as a model for additive noise. In particular, we plan to study the change of variables formula (Itô formula) for these integrals and their behaviour under change of measure. As an application we will try to derive an integral representation formula for generalized fractional Ornstein-Uhlenbeck processes, which are promising models for volatility in financial markets.

• DFG Programme: Research Grants