Weighted multilevel Langevin simulation of invariant measures. (English) Zbl 1404.60113

Summary: We investigate a weighted multilevel Richardson-Romberg extrapolation for the ergodic approximation of invariant distributions of diffusions adapted from the one introduced in [V. Lemaire and the first author, Bernoulli 23, No. 4A, 2643–2692 (2017; Zbl 1383.65003)] for regular Monte Carlo simulation. In a first result, we prove under weak confluence assumptions on the diffusion, that for any integer \(R\geq2\), the procedure allows us to attain a rate \(n^{\frac{R}{2R+1}}\) 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 \(\varepsilon>0\), 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 \(\varepsilon^{2}\) is about \(\varepsilon^{-2}\log(\varepsilon^{-1})\). { } 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.


60J60 Diffusion processes
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
65C05 Monte Carlo methods


Zbl 1383.65003
Full Text: DOI arXiv Euclid


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