Speaker: Prof Horst Osswald
(Mathematisches Institut der Universität München)

Title: Orthogonal polynomials, Chaos and Malliavin calculus for square integrable Lévy processes
 
Abstract:  Measures are considered, which are derived from the increments of square integrable Lévy-processes L (Brownian Motion and Poisson processes are examples). In order to develop Malliavin calculus for these measures, chaos decomposition of functionals in L2(µ)  is applied. To this end we use orthogonal polynomials of arbitrary finite degree in the increments of processes M which are very similar to L. We will show by an example that stochastic integration, where the integrands are Levy martingales Mt - E[Mt] of degree 1, is not sufficient: we need Lévy martingales of higher degree than 1.

The Malliavin derivative is a densely defined closed operator from L2(µ)  into the Hilbert space of two-parameter processes, dependent on time and the orthogonal polynomials. We study the Skorohod integral and we sketch the proof of a version of the Clark Ocone formula, which is a martingale representation of the functionals in L2(µ).