Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

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

by M. Gerencsér and A. Jentzen and D. Salimova

(Report number 2017-07)

Abstract
In the recent article [Jentzen, A., Müller-Gronbach, T., and Yaroslavtseva, L., Commun. Math. Sci., 14(6), 1477--1500, 2016] it has been established that for every arbitrarily slow convergence speed and every natural number \(d \in \{4,5,\ldots\}\) there exist \(d\)-dimensional stochastic differential equations (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. In this paper we strengthen the above result by proving that this slow convergence phenomena also arises in two (\(d=2\)) and three (\(d=3\)) space dimensions.

Keywords: Lower bounds, strong approximation, stochastic differential equation, SDE, numerical scheme, convergence rates

@Techreport{GJS17_703,
  author = {M. Gerencsér and A. Jentzen and D. Salimova},
  title = {On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-07},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-07.pdf },
  year = {2017}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).