  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.
