On the Asymptotic Moment Preservation of the Stochastic Theta Method for Multi-dimensional Ornstein-Uhlenbeck Processes

The preservation of long-term statistical properties is a key criterion for evaluating numerical method of stochastic differential equations. This paper gives the detailed and self-contained analysis of the stochastic theta method (STM) applied to multi-dimensional Ornstein-Uhlenbeck processes. Extending moment-preservation results from the one-dimensional to the multi-dimensional setting presents a major challenge, as it requires a shift from scalar analysis to an operator-theoretic framework based on continuous and discrete algebraic Lyapunov equations. Our main contribution lies in bridging this gap. We demonstrate that the STM preserves the asymptotic mean if and only if the method is mean-stable. This stability condition holds unconditionally for the implicit range of the parameter $\theta \in [1/2, 1]$, while a step-size restriction is required for parameter $\theta \in [0, 1/2)$. More importantly, by deriving the precise structure of the global error’s leading term, we prove that the trapezoidal scheme ($\theta=1/2$) is the only method in the STM family that achieves second-order accuracy for the asymptotic covariance matrix, while all other methods (STM with $\theta \neq 1/2$) exhibit only first-order accuracy.Finally, a two-dimensional colored noise model is included to verify our theoretical findings.