  We study a quite general class of stochastic dispersive equations with linear
multiplicative noise, including especially the Schr\"odinger and Airy
equations. The pathwise Strichartz and local smoothing estimates are derived
here in both the conservative and nonconservative case. In particular, we
obtain the P-integrability of constants in these estimates, where P is the
underlying probability measure. Several applications are given to nonlinear
problems, including local well-posedness of stochastic nonlinear Schr\"odinger
equations with variable coefficients and lower order perturbations,
integrability of global solutions to stochastic nonlinear Schr\"odinger
equations with constant coefficients. As another consequence, we prove as well
the large deviation principle for the small noise asymptotics.
