  It is well known that symplectic methods have been rigorously shown to be
superior to non-symplectic ones especially in long-time computation, when
applied to deterministic Hamiltonian systems. In this paper, we attempt to
study the probabilistic superiority of stochastic symplectic methods by means
of the theory of large deviations. We propose the concept of asymptotical
preservation of numerical methods for large deviations principles associated
with the exact solutions of the general stochastic Hamiltonian systems.
Concerning that the linear stochastic oscillator is one of the typical
stochastic Hamiltonian systems, we take it as the test equation in this paper
to obtain precise results about the rate functions of large deviations
principles for both exact and numerical solutions. Based on the
G\"artner--Ellis theorem, we first study the large deviations principles of the
mean position and the mean velocity for both the exact solution and its
numerical approximations. Then, we prove that stochastic symplectic methods
asymptotically preserve these two large deviations principles, but
non-symplectic ones do not. This indicates that stochastic symplectic methods
are able to approximate well the exponential decay speed of the "hitting
probability" of the mean position and mean velocity of the stochastic
oscillator. To the best of our knowledge, this is the first result about using
large deviations principle to show the superiority of stochastic symplectic
methods compared with non-symplectic ones in the existing literature.
