Model reduction for a class of singularly perturbed stochastic differential equations

A class of singularly perturbed stochastic differential equations (SDE) with linear drift and nonlinear diffusion terms is considered. We prove that, on a finite time interval, the trajectories of the slow variables can be well approximated by those of a system with reduced dimension as the singular perturbation parameter becomes small. In particular, we show that when this parameter becomes small the first and second moments of the reduced system´s variables closely approximate the first and second moments, respectively, of the slow variables of the singularly perturbed system. Chemical Langevin equations describing the stochastic dynamics of molecular systems with linear propensity functions including both fast and slow reactions fall within the class of SDEs considered here. We therefore illustrate the goodness of our approximation on a simulation example modeling a well known biomolecular system with fast and slow processes.