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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Optimal approximation of stochastic differential equations by adaptive step-size control
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by Norbert Hofmann, Thomas Müller-Gronbach and Klaus Ritter PDF
Math. Comp. 69 (2000), 1017-1034 Request permission

Abstract:

We study the pathwise (strong) approximation of scalar stochastic differential equations with respect to the global error in the $L_2$-norm. For equations with additive noise we establish a sharp lower error bound in the class of arbitrary methods that use a fixed number of observations of the driving Brownian motion. As a consequence, higher order methods do not exist if the global error is analyzed. We introduce an adaptive step-size control for the Euler scheme which performs asymptotically optimally. In particular, the new method is more efficient than an equidistant discretization. This superiority is confirmed in simulation experiments for equations with additive noise, as well as for general scalar equations.
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Additional Information
  • Norbert Hofmann
  • Affiliation: Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstraße 1 1/2, 91054 Erlangen, Germany
  • Email: hofmann@mi.uni-erlangen.de
  • Thomas Müller-Gronbach
  • Affiliation: Mathematisches Institut, Freie Universität Berlin, Arminallee 2–6, 14195 Berlin, Germany
  • Email: gronbach@math.fu-berlin.de
  • Klaus Ritter
  • Affiliation: Fakultät für Mathematik und Informatik, Universität Passau, Innstr. 33, 94032 Passau, Germany
  • Email: klaus.ritter@fmi.uni-passau.de
  • Received by editor(s): August 24, 1998
  • Published electronically: May 20, 1999
  • Additional Notes: The first author’s work was supported by the DFG:GR 876/9-2.
  • © Copyright 2000 American Mathematical Society
  • Journal: Math. Comp. 69 (2000), 1017-1034
  • MSC (1991): Primary 65U05; Secondary 60H10
  • DOI: https://doi.org/10.1090/S0025-5718-99-01177-1
  • MathSciNet review: 1677407