On the numerical structure preservation of nonlinear damped stochastic oscillators

The paper is focused on analyzing the conservation issues of stochastic theta-methods when applied to nonlinear damped stochastic oscillators. In particular, we are interested in reproducing the long-term properties of the continuous problem over its discretization through stochastic theta-methods, by preserving the correlation matrix. This evidence is equivalent to accurately maintaining the stationary density of the position and the velocity of a particle driven by a nonlinear deterministic forcing term and an additive noise as a stochastic forcing term. The provided analysis relies on a linearization of the nonlinear problem, whose effectiveness is proved theoretically and numerically confirmed.