D. Belomestny, L. Iosipoi † E. Moulines ‡, A. Naumov §, and S. Samsonov ¶ Abstract In this paper we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non asymptotic analysis of a variance reduced functional as well as by a thorough simulation study. In particular we apply our method to various MCMC Bayesian estimation problems where it favourably compares to the existing variance reduction approaches. 1 Introduction Variance reduction methods play nowadays a prominent role as a complexity reduction tool in simulation based numerical algorithms like Monte Carlo (MC) or Markov Chain Monte Carlo (MCMC).