Multilevel Richardson–Romberg extrapolation

Multilevel Monte Carlo methods

“…Lemaire and Pagès have taken this approach much further, and have also performed a comprehensive error analysis (Lemaire and Pagès 2013). Assuming that the weak error has a regular expansion…”

Numerical methods for Stochastic differential equations: two examples

“…Notice that the condition RC Q,V (I d , φ, α, β)(i) with φ concave appears in [4] to prove sub-geometrical ergodicity of Markov chains. In [12], a similar hypothesis to RC Q,V (I d , φ, α, β)(i), with φ not necessarily concave, is also used (with A γn replaced by A) to study the convergence of the weighted empirical measures (14) for the Euler scheme of a Brownian diffusion. The function φ controls the mean reverting property.…”

“…The function ψ is closely related to the identification of the set of test functions f for which we have lim n→+∞ ν η n (f ) = ν(f ) a.s., when ν is the unique invariant distribution of the underlying Feller process. To this end, for s 1, which is related to step weight assumption, we introduce the sets of test functions for which we will show the a.s. convergence of the weighted empirical measures (14):…”

“…This section presents the assumption that enables to characterize the limiting distributions of the a.s. tight sequence (ν η n (dx, ω)) n∈N * . We aim to estimate the distance between V and ν η n (see (14)) for n large enough. We thus introduce an hypothesis concerning the distance between ( A γ ) γ>0 , the pseudo-generator of (Q γ ) γ>0 , and A, the infinitesimal generator of (P t ) t 0 .…”

“…An extended version of the presented results, including detailed proofs, is available in [18]. In Section 4, is briefly presented an application of the multilevel (see [8]) paradigm (in its weighted version, see [14]), still for the recursive approximation of the invariant distribution in a Brownian diffusion framework. This is a joint work by G. Pagès and F. Panloup [17].…”