We analyse convergence of a micro-macro acceleration method for the MonteCarlo simulation of stochastic differential equations with time-scaleseparation between the (fast) evolution of individual trajectories and the(slow) evolution of the macroscopic function of interest. We consider a classof methods, presented in [Debrabant, K., Samaey, G., Zieli'nski, P. Amicro-macro acceleration method for the Monte Carlo simulation of stochasticdifferential equations. SINUM, 55 (2017) no. 6, 2745-2786], that performs shortbursts of path simulations, combined with the extrapolation of a fewmacroscopic state variables forward in time. After extrapolation, a newmicroscopic state is then constructed, consistent with the extrapolatedvariable and minimising the perturbation caused by the extrapolation. In thepresent paper, we study a specific method in which this perturbation isminimised in a relative entropy sense. We discuss why relative entropy is auseful metric, both from a theoretical and practical point of view, andrigorously study local errors and numerical stability of the resulting methodas a function of the extrapolation time step and the number of macroscopicstate variables. Using these results, we discuss convergence to the fullmicroscopic dynamics, in the limit when the extrapolation time step tends tozero and the number of macroscopic state variables tends to infinity.
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