Moment convergence in a class of singularly perturbed stochastic differential equations

We consider a class of singularly perturbed stochastic differential equations with linear drift and nonlinear diffusion terms. We obtain a reduced-order model that approximates the slow variable dynamics of the original system when the singular perturbation parameter e is small. In our previous work, it was shown that, on a finite time interval, the first and the second moments of the slow variable dynamics of the original system are within an O(ε)-neighborhood of the first and the second moments of the reduced-order system. In this paper, we extend this result to show that all moments of the slow variable dynamics of the original system are within an O(ε)-neighborhood of the moments of the reduced-order system. We illustrate the application of this approach on a biomolecular system modeled by the chemical Langevin equation.