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

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