Slow manifold and averaging for slow-fast stochastic differential system

We consider multiscale stochastic dynamical systems. In this article an intermediate reduced model is obtained for a slow-fast system with fast mode driven by white noise. First, the reduced stochastic system on exponentially attracting slow manifold reduced system is derived to errors of O(ǫ). Second, averaging derives an autonomous deterministic system up to errors of O(√ǫ). Then an intermediate reduced model, which is an autonomous deterministic system driven by white noise up to errors of O(ǫ), is derived using a martingale approach to account for fluctuations about the averaged system. This intermediate reduced model has a simpler form than the reduced model on the stochastic slow manifold. These results not only connect averaging with the slow manifold, they also provide a martingale method for improving averaged models of stochastic systems. MSC: 34C15; 37H10; 60H10