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December 2018 Weighted multilevel Langevin simulation of invariant measures
Gilles Pagès, Fabien Panloup
Ann. Appl. Probab. 28(6): 3358-3417 (December 2018). DOI: 10.1214/17-AAP1364

Abstract

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 $R\ge2$, 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.

Citation

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Gilles Pagès. Fabien Panloup. "Weighted multilevel Langevin simulation of invariant measures." Ann. Appl. Probab. 28 (6) 3358 - 3417, December 2018. https://doi.org/10.1214/17-AAP1364

Information

Received: 1 July 2016; Revised: 1 June 2017; Published: December 2018
First available in Project Euclid: 8 October 2018

zbMATH: 06994396
MathSciNet: MR3861816
Digital Object Identifier: 10.1214/17-AAP1364

Subjects:
Primary: 37M25 , 60J60 , 65C05

Keywords: Ergodic diffusion , ergodicity , invariant measure , Monte Carlo , multilevel , PAC-Bayesian , Richardson–Romberg

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 6 • December 2018
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