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In this paper, we derive a strong convergence rate of spatial finitedifference approximations for both focusing and defocusing stochastic cubicSchr\"odinger equations driven by a multiplicative $Q$-Wiener process. Beyondthe uniform boundedness of moments for high order derivatives of the exactsolution, the key requirement of our approach is the exponential integrabilityof both the exact and numerical solutions. By constructing and analyzing aLyapunov functional and its discrete correspondence, we derive the uniformboundedness of moments for high order derivatives of the exact solution and thefirst order derivative of the numerical solution, which immediately yields thewell-posedness of both the continuous and discrete problems. The latterexponential integrability is obtained through a variant of a criterion given by[Cox, Hutzenthaler and Jentzen, arXiv:1309.5595]. As a by-product of thisexponential integrability, we prove that the exact and numerical solutionsdepend continuously on the initial data and obtain a large deviation-typeresult on the dependence of the noise with first order strong convergence rate.

Strong Convergence Rate of Finite Difference Approximations for Stochastic Cubic Schr\"odinger Equations