Cylindrical martingale-valued measures, stochastic integration and stochastic PDEs in Hilbert space

A. E. Alvarado-Solano, C. A. Fonseca-Mora

Published 2020-01-24Version 1

We introduce a theory of stochastic integration for operator-valued integrands with respect to some classes of cylindrical martingale-valued measures in Hilbert spaces. The integral is constructed using a novel technique that utilizes the radonification of cylindrical martingales by a Hilbert-Schmidt operator theorem. We apply the developed theory of stochastic integration to establish existence and uniqueness of weak and mild solutions for stochastic evolution equations driven by multiplicative cylindrical martingale-valued measure noise with rather general coefficients. Our theory covers the study of integration and of SPDEs driven by Hilbert space valued L\'{e}vy noise (which is not required to satisfy any moment condition), cylindrical L\'{e}vy noise with (weak) second moments and L\'{e}vy-valued random martingale measures with finite second moment.