  In this paper, we derive a strong convergence rate of spatial finite
difference approximations for both focusing and defocusing stochastic cubic
Schr\"odinger equations driven by a multiplicative $Q$-Wiener process. Beyond
the uniform boundedness of moments for high order derivatives of the exact
solution, the key requirement of our approach is the exponential integrability
of both the exact and numerical solutions. By constructing and analyzing a
Lyapunov functional and its discrete correspondence, we derive the uniform
boundedness of moments for high order derivatives of the exact solution and the
first order derivative of the numerical solution, which immediately yields the
well-posedness of both the continuous and discrete problems. The latter
exponential integrability is obtained through a variant of a criterion given by
[Cox, Hutzenthaler and Jentzen, arXiv:1309.5595]. As a by-product of this
exponential integrability, we prove that the exact and numerical solutions
depend continuously on the initial data and obtain a large deviation-type
result on the dependence of the noise with first order strong convergence rate.

Probability