Weighted Multilevel Langevin Simulation of Invariant Measures

TL;DR

This paper introduces a weighted multilevel Langevin simulation method that significantly improves the convergence rate for approximating invariant measures of diffusions, without increasing variance, and demonstrates its efficiency through numerical experiments.

Contribution

It develops a multilevel Richardson-Romberg extrapolation technique for ergodic diffusion approximation, achieving faster convergence rates and optimized parameters under certain assumptions.

Findings

Achieves convergence rate of n^{R/(2R+1)} for any R≥2

Maintains low asymptotic variance despite faster convergence

Numerical experiments confirm theoretical efficiency on various examples

Abstract

We investigate a weighted Multilevel Richardson-Romberg extrapolation for the ergodic approximation of invariant distributions of diffusions adapted from the one introduced in~[Lemaire-Pag\`es, 2013] 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 $\varepsilon\textgreater{}0$, 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 this Multilevel Langevin estimator on several examples including the simple one-dimensional Ornstein-Uhlenbeck process but also on a high dimensional diffusion motivated by a statistical problem. These examples confirm the theoretical efficiency of the method.