Variance-reduced HMM for Stochastic Slow-fast Systems

We propose a novel variance reduction strategy based on control variables for simulating the averaged equation of a stochastic slow-fast system. In this system, we assume that the fast equation is ergodic, implying the existence of an invariant measure, for every fixed value of the slow variable. The right hand side of the averaged equation contains an integral with respect to this unknown invariant measure, which is approximated by the heterogeneous multiscale method (HMM). The HMM method corresponds to a Markov chain Monte Carlo method in which samples are generated by simulating the fast equation for a fixed value of the slow variable. As a consequence, the variance of the HMM estimator decays slowly. Here, we introduce a variance-reduced HMM estimator based on control variables: from the current time HMM estimation, we subtract a second HMM estimator at the previous time step using the same seed as the current time HMM estimator. To avoid introducing a bias, we add the previously calculated variance-reduced estimator. We analyze convergence of the proposed estimator and apply it to a linear and nonlinear model