## 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.

### MSC:

 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
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### References:

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