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A Variable Step Size Riemannian Sum for an Itô Integral

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

E. Rapoo
E. Rapoo*
Affiliation:
University of South Africa
*
∗Postal address: Department of Mathematical Sciences, University of South Africa, PO Box 392, UNISA 0003, Republic of South Africa. Email address: rapooe@unisa.ac.za
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Abstract

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We investigate the problem of using a Riemannian sum with random subintervals to approximate the iterated Itô integral ∫wdw - or, equivalently, solving the corresponding stochastic differential equation by Euler's method with variable step sizes. In the past this task has been used as a counterexample to illustrate that variable step sizes must be used with extreme caution in stochastic numerical analysis. This article establishes a class of variable step size schemes which do work.

Keywords

Adaptive time discretisationvariable step sizenumerical simulationpathwise approximationstochastic differential equationmean-square numerical method

MSC classification

Secondary: 65C20: Models, numerical methods 65C30: Stochastic differential and integral equations 60H05: Stochastic integrals
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 551 - 567
Copyright
Copyright © Applied Probability Trust 2008 

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