A relatively short proof of itô’s formula for spdes and its applications

Abstract

We give a short proof of Itô’s formula for stochastic Hilbert-space valued processes in the setting V ⊂ H ⊂ V∗ based on the possibility to lift the stochastic differentials, which are originally in V∗, into H. Using this result we also prove the maximum principle for second-order SPDEs in arbitrary domains.