Variance reduction for additive functionals of Markov chains via martingale representations

In this paper, we propose an efficient variance reduction approach for additive functionals of Markov chains relying on a novel discrete-time martingale representation. Our approach is fully non-asymptotic and does not require the knowledge of the stationary distribution (and even any type of ergodicity) or specific structure of the underlying density. By rigorously analyzing the convergence properties of the proposed algorithm, we show that its cost-to-variance product is indeed smaller than one of the naive algorithms. The numerical performance of the new method is illustrated for the Langevin-type Markov chain Monte Carlo (MCMC) methods.