Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces: An L2 approach

Abstract

In this work stochastic integration with respect to cylindrical Lévy processes with weak second moments is introduced. It is well known that a deterministic Hilbert–Schmidt operator radonifies a cylindrical random variable, i.e. it maps a cylindrical random variable to a classical Hilbert space valued random variable. Our approach is based on a generalisation of this result to the radonification of the cylindrical increments of a cylindrical Lévy process by random Hilbert–Schmidt operators. This generalisation enables us to introduce a Hilbert space valued random variable as the stochastic integral of a predictable stochastic process with respect to a cylindrical Lévy process. We finish this work by deriving an Itô isometry and by considering shortly stochastic partial differential equations driven by cylindrical Lévy processes.