Convergence of a $θ$-scheme to solve the stochastic nonlinear Schrödinger equation with Stratonovich noise

We propose a $θ$-scheme to discretize the $d$-dimensional stochastic cubic Schrödinger 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.