The Annals of Applied Probability

Step size control for the uniform approximation of systems of stochastic differential equations with additive noise

Norbert Hofmann, Thomas Müller-Gronbach, and Klaus Ritter

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Abstract

We analyze the pathwise approximation for systems of stochastic differential equations.The pathwise distance between the solution and its approximation is measured globally on the unit interval in the $L_{\infty}$-norm, and we study the expectation of this distance. For systems with additive noise we obtain sharp lower and upper bounds for the minimal error in the class of arbitrary methods which use discrete observations of a Brownian path. The optimal order is achieved by an Euler scheme with adaptive step-size control. We illustrate the superiority of the adaptive method compared to equidistant discretization by a simulation experiment.

Article information

Source
Ann. Appl. Probab., Volume 10, Number 2 (2000), 616-633.

Dates
First available in Project Euclid: 22 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1019487358

Digital Object Identifier
doi:10.1214/aoap/1019487358

Mathematical Reviews number (MathSciNet)
MR1768220

Zentralblatt MATH identifier
1054.65007

Subjects
Primary: 65U05
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Systems of stochastic differential equations pathwise approximation adaption step-size control asymptotic optimality

Citation

Hofmann, Norbert; Müller-Gronbach, Thomas; Ritter, Klaus. Step size control for the uniform approximation of systems of stochastic differential equations with additive noise. Ann. Appl. Probab. 10 (2000), no. 2, 616--633. doi:10.1214/aoap/1019487358. https://projecteuclid.org/euclid.aoap/1019487358


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The Annals of Applied Probability

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