Convergence of a $\theta$-scheme to solve the stochastic nonlinear Schr\"odinger equation with Stratonovich noise

We propose a $\theta$-scheme to discretize the $d$-dimensional stochastic cubic Schr"odinger equation in Stratono-vich sense. A uniform bound for the Hamiltonian of the discrete problem is obtained, which is a crucial property to verify the convergence in probability towards a mild solution. Furthermore, based on the uniform bounds of iterates in ${\mathbb{H}}^2(\mathcal{O})$ for $\mathcal{O}\subset {\mathbb{R}}^1$, the optimal convergence order 1 in strong local sense is obtained.