Continuity properties of Hilbert space valued martingales

Necessary and sufficient conditions for Hölder continuity of Hilbert space valued martingales are given in terms of the associated quadratic variation. As an application one obtains a sufficient condition for a mild solution of a stochastic evolution equation to have a continuous version if the semigroup governing this equation is analytic. Further we derive Levy's modulus of continuity for the Hilbert space valued stochastic integral with the Wiener process as integrator and obtain a generalization of the loglog law for that integral.