The nonlinear stochastic Schr\"odinger equation via stochastic Strichartz estimates

We consider the stochastic NLS with nonlinear Stratonovic noise for initial values in $L^2(R^d)$ and prove local existence and uniqueness of a mild solution for subcritical and critical nonlinearities. The proof is based on deterministic and stochastic Strichartz estimates. In the subcritical case, we prove that the solution is global, if we impose an additional assumption on the nonlinear noise.