The quantitative characterization of the evolution of the error distribution (as the step-size tends to zero) is a fundamental problem in the analysis of stochastic numerical method. In this paper, we answer this problem by proving that the error of numerical method for linear stochastic differential equation satisfies the limit theorems and large deviation principle. To the best of our knowledge, this is the first result on the quantitative characterization of the evolution of the error distribution of stochastic numerical method. As an application, we provide a new perspective to explain the superiority of symplectic methods for stochastic Hamiltonian systems in the long-time computation. To be specific, by taking the linear stochastic oscillator as the test equation, we show that in the long-time computation, the probability that the error deviates from the typical value is smaller for the symplectic methods than that for the non-symplectic methods, which reveals that the stochastic symplectic methods are more stable than non-symplectic methods.

翻译：误差分布随步长趋近于零的定量刻画是随机数值方法分析中的基本问题。本文通过证明线性随机微分方程数值方法的误差满足极限定理和大偏差原理，回答了该问题。据我们所知，这是关于随机数值方法误差分布定量演化的首个结果。作为应用，我们为随机哈密顿系统中辛方法在长时间计算中的优越性提供了新的解释视角。具体而言，以线性随机振子为测试方程，我们证明在长时间计算中，辛方法的误差偏离典型值的概率小于非辛方法，这揭示了随机辛方法比非辛方法具有更高的稳定性。