  We analyse convergence of a micro-macro acceleration method for the Monte
Carlo simulation of stochastic differential equations with time-scale
separation between the (fast) evolution of individual trajectories and the
(slow) evolution of the macroscopic function of interest. We consider a class
of methods, presented in [Debrabant, K., Samaey, G., Zieli\'nski, P. A
micro-macro acceleration method for the Monte Carlo simulation of stochastic
differential equations. SINUM, 55 (2017) no. 6, 2745-2786], that performs short
bursts of path simulations, combined with the extrapolation of a few
macroscopic state variables forward in time. After extrapolation, a new
microscopic state is then constructed, consistent with the extrapolated
variable and minimising the perturbation caused by the extrapolation. In the
present paper, we study a specific method in which this perturbation is
minimised in a relative entropy sense. We discuss why relative entropy is a
useful metric, both from a theoretical and practical point of view, and
rigorously study local errors and numerical stability of the resulting method
as a function of the extrapolation time step and the number of macroscopic
state variables. Using these results, we discuss convergence to the full
microscopic dynamics, in the limit when the extrapolation time step tends to
zero and the number of macroscopic state variables tends to infinity.
