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Optimal approximation
of stochastic differential equations
by adaptive step-size control


Authors: Norbert Hofmann, Thomas Müller-Gronbach and Klaus Ritter
Journal: Math. Comp. 69 (2000), 1017-1034
MSC (1991): Primary 65U05; Secondary 60H10
DOI: https://doi.org/10.1090/S0025-5718-99-01177-1
Published electronically: May 20, 1999
MathSciNet review: 1677407
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Abstract | References | Similar Articles | Additional Information

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

DOI: https://doi.org/10.1090/S0025-5718-99-01177-1
Keywords: Stochastic differential equations, pathwise approximation, adaption, step-size control, asymptotic optimality
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.
Article copyright: © Copyright 2000 American Mathematical Society