Exponentially Fitted Midpoint Scheme for a Stochastic Oscillator

In this paper, we propose the exponentially fitted midpoint scheme for the stochastic oscillator. This scheme is first-order strongly convergent and it preserves symplectic. It can effectively simulate the oscillatory behavior of stochastic oscillators, and its second moment grows linearly with time. In addition, we also propose a two-parameter estimation method by analyzing the expectation and variance in the discrete scheme. Numerical experiments are given to verify effectiveness of the exponential fitting method and parameter estimation methods based on this scheme.