WEIGHTED MULTILEVEL LANGEVIN SIMULATION OF INVARIANT MEASURES

We investigate a weighted multilevel Richardson–Romberg extrapolation for the ergodic approximation of invariant distributions of diffusions adapted from the one introduced in [Bernoulli 23 (2017) 2643–2692] for regular Monte Carlo simulation. In a first result, we prove under weak confluence assumptions on the diffusion, that for any integer R ≥ 2, the procedure allows us to attain a rate R n 2 R + 1 whereas the original algorithm convergence is at a weak rate n 1/3. Furthermore, this is achieved without any explosion of the asymptotic variance. In a second part, under stronger confluence assumptions and with the help of some second-order expansions of the asymptotic error, we go deeper in the study by optimizing the choice of the parameters involved by the method. In particular, for a given ε >0, we exhibit some semi-explicit parameters for which the number of iterations of the Euler scheme required to attain a mean-squared error lower than ε² is about ε −2 log(ε −1). Finally, we numerically test this multilevel Langevin estimator on several examples including the simple one-dimensional Ornstein–Uhlenbeck process but also a high dimensional diffusion motivated by a statistical problem. These examples confirm the theoretical efficiency of the method.