  In this paper, we prove the well-posedness and op- timal trajectory
regularity for the solution of stochastic evolution equations driven by general
multiplicative noises in martingale type 2 Banach spaces. The main idea of our
method is to combine the approach in [HL] dealing with Hilbert setting and a
version of Burkholder inequality in M-type 2 Banach space. Applying our main
results to the stochastic heat equation gives a positive an- swer to an open
problem proposed in [JR12].
