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

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