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The Pathwise Convergence of Approximation Schemes for Stochastic Differential Equations

Published online by Cambridge University Press:  01 February 2010

P.E. Kloeden
Affiliation:
Johann Wolfgang Goethe-Universität, Institut für Mathematik, Robert-Mayer-Strasse 10, 60325 Frankfurt am Main, Germany, [email protected], http://www.math.uni-frankfurt.de/~numerik/kloeden/
A. Neuenkirch
Affiliation:
Johann Wolfgang Goethe-Universität, Institut für Mathematik, Robert-Mayer-Strasse 10, 60325 Frankfurt am Main, Germany, [email protected], http://www.math.uni-frankfurt.de/~neuenkir/

Abstract

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The authors of this paper study approximation methods for stochastic differential equations, and point out a simple relation between the order of convergence in the pth mean and the order of convergence in the pathwise sense: Convergence in the pth mean of order α for all p ≥ 1 implies pathwise convergence of order α – ε for arbitrary ε > 0. The authors then apply this result to several one-step and multi-step approximation schemes for stochastic differential equations and stochastic delay differential equations. In addition, they give some numerical examples.

Type
Research Article
Copyright
Copyright © London Mathematical Society 2007

References

1.Baker, C.T.H. and Buckwar, E., ‘Numerical analysis of explicit one-step methods for stochastic delay differential equations’, LMS J. Comput. Math. 3 (2000) 315335, http://www.lms.ac.Uk/jcm/3/lms2000-002.CrossRefGoogle Scholar
2.Brugnano, L., Burrage, K. and Burrage, P. M., ‘Adams-type methods for the numerical solution of stochastic ordinary differential equations’, BIT 40 (2000) 451470.CrossRefGoogle Scholar
3.Buckwar, E. and Winkler, R., ‘Multistep methods for SDEs and their application to problems with small noise’, SIAM J. Numer. Anal. 44 (2006) 779803CrossRefGoogle Scholar
4.Grüne, L. and Kloeden, P. E., ‘Pathwise approximation of random ordinary differential equations‘, BIT 41 (2001) 711721CrossRefGoogle Scholar
5.GYöNGY, I., ‘A note on Euler’s approximations’, Potential Anal. 8 (1998)205216CrossRefGoogle Scholar
6.Hairer, E., Nørset, S. P. and Wanner, G., Solving ordinary differential equations. I: non stiff problems, 2nd edn (Springer, Berlin, 1993).Google Scholar
7.Hu, Y., Mohammed, S.-E. A. and YAN, F., ‘Discrete-time approximations of stochastic delay equations: the Milstein scheme’, Ann. Probab. 32 (2004) 265314CrossRefGoogle Scholar
8.Jacod, J. and Protter, P., ‘Asymptotic error distributions for the Euler method for stochastic differential equations’, Ann. Prob. 26 (1998) 267307CrossRefGoogle Scholar
9.Kloeden, P. E. and Platen, E., Numerical solution of stochastic differential equations, 3rd edn (Springer, Berlin, 1999).Google Scholar
10.Mao, X., Stochastic differential equations and their applications(HorwoodPublishing Ltd, Chichester, 1997).Google Scholar
11.Mao, X., ‘Numerical solutions of stochastic functional differential equations’, LMS J. Comput. Math. 6 (2003) 141161, http://www.lms.ac.uk/jcm/6/lms2002-027.CrossRefGoogle Scholar
12.Milstein, G. N., Numerical integration of stochastic differential equations (Kluwer, Doordrecht, 1995).CrossRefGoogle Scholar
13.Neuenkirch, A. and Nourdin, I., ‘Exact rate of convergence of some approximation schemes associated to SDEs driven by a fBm’, J. Theor. Probab. (2007), to appear.CrossRefGoogle Scholar
14.Nualart, D., The Malliavin calculus and related topics, 2nd edn (Springer, Berlin, 1995).CrossRefGoogle Scholar
15.Römish, W. and Winkler, R., ‘Stepsize control for mean-square numerical methods for stochastic differential equations with small noise’, SIAM J. Sci. Comput. 28 (2006) 604625.CrossRefGoogle Scholar
16.Talay, D., ‘Résolution trajectorielle et analyse numérique des équations différentielles stochastiques’, Stochastics 9 (1983) 275306CrossRefGoogle Scholar
17.Yan, L., ‘Asymptoic error for the Milstein scheme for SDE driven by continous semimartingales’, Ann. Appl. probab. 15 (2005) 27062738CrossRefGoogle Scholar