On Some non Asymptotic Bounds for the Euler Scheme

By Vincent Lemaire, Stéphane Menozzi. 2010

66 citations recorded by the source corpus.

We obtain non asymptotic bounds for the Monte Carlo algorithm associated to the Euler discretization of some diffusion processes. The key tool is the Gaussian concentration satisfied by the density of the discretization scheme. This Gaussian concentration is derived from a Gaussian upper bound of the density of the scheme and a modification of the so-called "Herbst argument" used to prove Logarithmic Sobolev inequalities. We eventually establish a Gaussian lower bound for the density of the scheme that emphasizes the concentration is sharp.

Subfields: Applied Mathematics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3d91ffd2b149d0fd6b04abefb064796b65016407

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In Applied Mathematics

In Numerical Analysis

In Statistics and Probability

Citations in the corpus

Strong convergence for the Euler-Maruyama approximation of stochastic differential equations with discontinuous coefficients (2016, methodology): “establishthe strong rateof convergence of EMapproximation for SDEs with discontinuous drift are either the Krylov estimate (see [13, 6]) or theGaussianboundestimateforthedensityofthenumericalsolution([15,20,22]). However, these estimates seem no longer available for SDEs with discontinuous diffusion”
Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm (2015, background): “Compared to the related works by [23], [24], [25] and [26], we establish not only the weak convergence of the weighted empirical measure of the path to the target distribution but a much stronger convergence in total variation, similarly to [12], where the strongly log-concave case is considered.”