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In this work we introduce a theory of stochastic integration foroperator-valued integrands with respect to some classes of cylindricalmartingale-valued measures in Hilbert spaces. The integral is constructed viathe radonification of cylindrical martingales by a Hilbert-Schmidt operatortheorem and unifies several other theories of stochastic integration in Hilbertspaces. In particular, our theory covers the theory of stochastic integrationwith respect to a Hilbert space valued L\'{e}vy process (which is not requiredto satisfy any moment condition), with respect to a cylindrical L\'{e}vyprocesses with (weak) second moments and with respect to a L\'{e}vy-valuedrandom martingale measures with finite second moment. As an application of ourtheory of integration we prove existence and uniqueness of solutions forstochastic stochastic partial differential equations driven by multiplicativecylindrical martingale-valued measure noise with rather general coefficients.

Stochastic integration in Hilbert spaces with respect to cylindrical martingale-valued measures