Model reduction for a class of singularly perturbed stochastic differential equations: Fast variable approximation

Abstract

We consider a class of stochastic differential equations in singular perturbation form, where the drift terms are linear and diffusion terms are nonlinear functions of the state variables. In our previous work, we approximated the slow variable dynamics of the original system by a reduced-order model when the singular perturbation parameter ϵ is small. In this work, we obtain an approximation for the fast variable dynamics. We prove that the first and second moments of the approximation are within an O(ϵ)-neighborhood of the first and second moments of the fast variable of the original system. The result holds for a finite time-interval after an initial transient has elapsed. We illustrate our results with a biomolecular system modeled by the chemical Langevin equation.