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In this article we address the stability of linear stochastic differential equations. In particular, we focus our attention on non-normality in stochastic differential equations. Following Higham and Mao we study a test problem for non-normal stochastic differential equations, that is stable without noise, and prove a property conjectured by Higham and Mao, that is that an exponentially small (in the dimension) noise term is able to destabilise in a mean-square sense the solution of the SDE. [ABSTRACT FROM AUTHOR]

The paper is focused on analyzing the conservation issues of stochastic 𝜃-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 𝜃-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. [ABSTRACT FROM AUTHOR]

We introduce a theory of two-step Runge-Kutta (TSRK) methods for stochastic differential equations, arising from the perturbation of the corresponding TSRK methods for deterministic problems. We present a proof of convergence and study the mean-square stability properties. Numerical experiments confirming the theoretical results are provided. [ABSTRACT FROM AUTHOR]