Communications in Mathematical Sciences

Volume 14 (2016)

Number 6

On stochastic differential equations with arbitrary slow convergence rates for strong approximation

Pages: 1477 – 1500

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n6.a1

Authors

Arnulf Jentzen (Seminar für Angewandte Mathematik,, Departement Mathematik, ETH Zürich, Switzerland)

Thomas Müller-Gronbach (Fakultät für Informatik und Mathematik, Universität Passau, Germany)

Larisa Yaroslavtseva (Fakultät für Informatik und Mathematik, Universität Passau, Germany)

Abstract

In the recent article [M. Hairer, M. Hutzenthaler, and A. Jentzen, Ann. Probab., 43(2), 468–527, 2015] it has been shown that there exist stochastic differential equations (SDEs) with infinitely often differentiable and globally bounded coefficients such that the Euler scheme converges to the solution in the strong sense but with no polynomial rate. The result of Hairer et al. naturally leads to the question whether this slow convergence phenomenon can be overcome by using a more sophisticated approximation method than the simple Euler scheme. In this article we answer this question to the negative. We prove that there exist SDEs with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion converges in absolute mean to the solution with a polynomial rate. Even worse, we prove that for every arbitrarily slow convergence speed there exist SDEs with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence.

Keywords

stochastic differential equation, smooth coefficients, strong approximation, lower error bounds, slow convergence rate

2010 Mathematics Subject Classification

60H35, 65C30

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Published 12 August 2016