In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schrödinger 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.

Strong and weak approximation errors of a spatial finite element method are analyzed for stochastic partial differential equations(SPDEs) with one-sided Lipschitz coefficients, including the stochastic Allen-Cahn equation, driven by additive noise. In order to give the strong convergence rate of the finite element method, we present an appropriate decomposition and some a priori estimates of the discrete stochastic convolution. To the best of our knowledge, there has been no essentially sharp weak convergence rate of spatial approximation for parabolic SPDEs with nonglobally Lipschitz coefficients. To investigate the weak error, we first regularize the original equation by the splitting technique and derive the regularity estimates of the corresponding regularized Kolmogorov equation. Meanwhile, we present the regularity estimate in Malliavin sense and the refined estimate of the finite element method. Combining the regularity estimates of regularized Kolmogorov equation with Malliavin integration by parts formula, the weak convergence rate is shown to be twice the strong convergence rate.

This article analyzes an explicit temporal splitting numerical scheme for the stochastic Allen-Cahn equation driven by additive noise, in a bounded spatial domain with smooth boundary in dimension d ≤ 3. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem.When d = 1 and the driving noise is a space-time white noise, we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity, we then prove that under very mild assumptions on the initial data, this scheme achieves the optimal strong convergence rate O(δt 1 4 ). When d ≤ 3 and the driving noise possesses some regularity in space, we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension d = 1, these properties are used to prove that the splitting scheme has a strong convergence rate O(δt).2010 Mathematics Subject Classification. Primary 60H35; Secondary 60H15, 65M15.