Preprint No. A-98-09

Norbert Hofmann, Thomas Müller-Gronbach, Klaus Ritter

Optimal Approximation of Stochastic Differential Equations by Adaptive Step-Size Control

Abstract: We study the pathwise (strong) approximation of scalar stochastic differential equations with respect to the global error in the $L_2$-norm. We introduce an adaptive step-size control for the Euler scheme. For equations with additive noise this method is asymptotically optimal in the class of arbitrary methods that use a fixed number of observations of the driving Brownian motion. The superiority of the new method is confirmed in simulation experiments for equations with additive noise as well as for general scalar equations.

Keywords: Stochastic differential equations, pathwise approximation, step-size control, asymptotic optimality

Mathematics Subject Classification (MSC91): 65U05 Numerical methods in probability and statistics, 60H10 Stochastic ordinary differential equations [See also 34F05]

Language: ENG


Contact: Müller-Gronbach, Thomas; Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (

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