A decreasing step method for strongly oscillating stochastic models

We propose an algorithm for approximating the solution of a strongly oscillating SDE, that is, a system in which some ergodic state variables evolve quickly with respect to the other variables. The algorithm profits from homogenization results and consists of an Euler scheme for the slow scale variables coupled with a decreasing step estimator for the ergodic averages of the quick variables. We prove the strong convergence of the algorithm as well as a C.L.T. like limit result for the normalized error distribution. In addition, we propose an extrapolated version that has an asymptotically lower complexity and satisfies the same properties as the original version.