On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions Journal Article


Author(s): Gerencsér, Máté; Jentzen, Arnulf; Salimova, Diyora
Article Title: On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions
Affiliation IST Austria
Abstract: In a recent article (Jentzen et al. 2016 Commun. Math. Sci. 14, 1477–1500 (doi:10.4310/CMS.2016.v14. n6.a1)), it has been established that, for every arbitrarily slow convergence speed and every natural number d ? {4, 5, . . .}, there exist d-dimensional stochastic differential equations 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. In this paper, we strengthen the above result by proving that this slow convergence phenomenon also arises in two (d = 2) and three (d = 3) space dimensions.
Keywords: Lower error bounds; slow convergence rate; smooth coefficients; stochastic differential equation
Journal Title: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume: 473
Issue 2207
ISSN: 1364-5021
Publisher: Royal Society of London  
Date Published: 2017-11-01
Start Page: Article number: 20170104
URL:
DOI: 10.1098/rspa.2017.0104
Open access: yes (repository)
IST Austria Authors