  We consider a periodic nonlinear Schr\"odinger equation with white noise
dispersion and a power nonlinearity given by
  \begin{equation*}
  idu = \Delta u \circ dW_t + |u|^{p-1}u\;dt
  \end{equation*}
  By proving stochastic Strichartz estimates, we are able to prove almost sure
global wellposedness of this equation with $L^2$ initial data for
nonlinearities with exponent $1 < p \leq 3$. By generalizing the Fourier
restriction spaces $X^{s,b}$ to the stochastic setting, we also prove that our
solutions agree with the ones constructed by Chouk and Gubinelli using rough
path techniques. We also consider the quintic equation ($p=5$), and show that
it is analytically illposed in $L^1_\omega C_t L^2_x$.
