Markus Riedle (KCL)

Stochastic integration in Banach spaces and radonifying operators

Abstract:
One of the cores in modern probability theory is the stochastic integral introduced by K. Ito in the 1940s. Due to the randomness and the irregularity of typical stochastic integrators (such as the Wiener process) one can not follow a classical approach as in calculus to define the stochastic integral. In Hilbert spaces stochastic integration with respect to Wiener processes can be introduced for Hilbert-Schmidt operators as integrands whereas the recently developed generalisation to Banach spaces requires $\gamma$-radonifying operators as integrands. In this talk we explain the relation between the extensively studied class of $\gamma$-radonifying operators and stochastic integration with respect to Wiener processes. Surprisingly, it turns out that for more general integrators which are non-Gaussian and discontinuous (Levy processes) such a relation can still be established but with another subclass of radonifying operators.