arXiv subjects

Oumaima Bencheikh

Publications and source records attributed to Oumaima Bencheikh.

4 recordsLinked to original sources

arXiv2021

Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures

We are interested in the approximation in Wasserstein distance with index $ρ\ge 1$ of a probability measure $μ$ on the real line with finite moment of order $ρ$ by the empirical measure of $N$ deterministic points. The minimal error converges to $0$ as $N\to+\infty$ and we try to characterize the order associated with this convergence. In \cite{xuberger}, Xu and Berger show that, apart when $μ$ is a Dirac mass and the error vanishes, the order is not larger than $1$ and give a sufficient condition for the order to be equal to this threshold $1$ in terms of the density of the absolutely continuous with respect to the Lebesgue measure part of $μ$. They also prove that the order is not smaller than $1/ρ$ when the support of $μ$ is bounded and not larger when the support is not an interval. We complement these results by checking that for the order to lie in the interval $\left(1/ρ,1\right)$, the support has to be bounded and by stating a necessary and sufficient condition in terms of the tails of $μ$ for the order to be equal to some given value in the interval $\left(0,1/ρ\right)$, thus precising the sufficient condition in terms of moments given in \cite{xuberger}. In view of practical application, we emphasize that in the proof of each result about the order of convergence of the minimal error, we exhibit a choice of points explicit in terms of the quantile function of $μ$ which exhibits the same order of convergence.

math.PR↗

arXiv2020

Weak and strong error analysis for mean-field rank based particle approximations of one dimensional viscous scalar conservation law

In this paper, we analyse the rate of convergence of a system of $N$ interacting particles with mean-field rank based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikhov to check trajectorial propagation of chaos with optimal rate $N^{-1/2}$ to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy to check convergence in $L^1(\mathbb{R})$ with rate ${\mathcal O}(\frac{1}{\sqrt N} + h)$ of the empirical cumulative distribution function of the Euler discretization with step $h$ of the particle system to the solution of a one dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves in ${\mathcal O}(\frac{1}{N} + h)$. We provide numerical results which confirm our theoretical estimates.

math.PR↗

arXiv2020

Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise

We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order $1/2$ in total variation distance. When the drift has a spatial divergence in the sense of distributions with $ρ$-th power integrable with respect to the Lebesgue measure in space uniformly in time for some $ρ\ge d$, the order of convergence at the terminal time improves to $1$ up to some logarithmic factor. In dimension $d=1$, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. We confirm our theoretical analysis by numerical experiments.

math.PR↗

arXiv2018

Bias behaviour and antithetic sampling in mean-field particle approximations of SDEs nonlinear in the sense of McKean

In this paper, we prove that the weak error between a stochastic differential equation with nonlinearity in the sense of McKean given by moments and its approximation by the Euler discretization with time-step h of a system of N interacting particles is O(1/N + h). We provide numerical experiments confirming this behaviour and showing that it extends to more general mean-field interaction and study the efficiency of the antithetic sampling technique on the same examples.

math.PR↗