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In this work stochastic integration with respect to cylindrical Levy 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 Levy 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 Levy process. We finish this work by deriving an Ito isometry and by considering shortly stochastic partial differential equations driven by cylindrical Levy processes.

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

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