|Universiteit Faculteit FNWI English version||15 april 2010, e-mail|
Woensdag 21 april 2010
AbstractA succesful analysis of stochastic differential equations requires a suitable notion of stochastic integral. For stochastic differential equations in Hilbert spaces driven by Wiener processes an isometry due to Ito yields a convenient stochastic integral and a rich theory has been developed. The situation is quite different for equations in Banach spaces, since an Ito isometry does in general not hold for Banach space valued functions.
Van Neerven and Weis define a Banach space valued stochastic integral for Wiener processes in a Pettis sense, that is, by applying functionals. Using theory of Gaussian measures, they show that stochastic integrability of a Banach space valued function is equivalent to an associated integral operator being gamma-radonifying. They also derive an isometry involving a suitable norm on the space of gamma-radonifying operators.
A next step is to replace the Wiener process by a more general Levy process. Although the distributions of Levy processes are in general not Gaussian and their paths may be discontinuous, the ideas of Van Neerven and Weis can be modified to build a stochastic integration theory for Banach space valued functions with respect to arbitrary Levy processes. The ensuing stochastic integral can be used to solve corresponding stochastic differential equations and to prove a Levy-Ito decomposition for Banach space valued Levy processes.