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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
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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  • 1. Nicolas Bouleau and Dominique Lépingle, Numerical methods for stochastic processes, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1994. A Wiley-Interscience Publication. MR 1274043
  • 2. Stamatis Cambanis and Yaozhong Hu, Exact convergence rate of the Euler-Maruyama scheme, with application to sampling design, Stochastics Stochastics Rep. 59 (1996), no. 3-4, 211–240. MR 1427739
  • 3. J. M. C. Clark and R. J. Cameron, The maximum rate of convergence of discrete approximations for stochastic differential equations, Stochastic differential systems (Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978) Lecture Notes in Control and Information Sci., vol. 25, Springer, Berlin-New York, 1980, pp. 162–171. MR 609181
  • 4. Faure, O. (1992). Simulation du mouvement brownien et des diffusions. Thèse ENPC, Paris.
  • 5. J. G. Gaines and T. J. Lyons, Variable step size control in the numerical solution of stochastic differential equations, SIAM J. Appl. Math. 57 (1997), no. 5, 1455–1484. MR 1470933,
  • 6. Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR 1214374
  • 7. V. E. Maiorov, About widths of Wiener space in the 𝐿_{𝑞}-norm, J. Complexity 12 (1996), no. 1, 47–57. MR 1386593,
  • 8. Mauthner, S. (1998). Step size control in the numerical solution of stochastic differential equations. Mathematische Preprintreihe der TU Darmstadt 1972; finally published in J. Comput. Appl. Math. 100 (1998), 93-109. CMP 99:05
  • 9. G. N. Milstein, Numerical integration of stochastic differential equations, Mathematics and its Applications, vol. 313, Kluwer Academic Publishers Group, Dordrecht, 1995. Translated and revised from the 1988 Russian original. MR 1335454
  • 10. Thomas Müller-Gronbach, Optimal designs for approximating the path of a stochastic process, J. Statist. Plann. Inference 49 (1996), no. 3, 371–385. MR 1381165,
  • 11. Nigel J. Newton, An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times, Stochastics Stochastics Rep. 29 (1990), no. 2, 227–258. MR 1041038
  • 12. Erich Novak and Klaus Ritter, Some complexity results for zero finding for univariate functions, J. Complexity 9 (1993), no. 1, 15–40. Festschrift for Joseph F. Traub, Part I. MR 1213485,
  • 13. Ritter, K. (1999). Average Case Analysis of Numerical Problems. Lect. Notes in Math., Springer, Berlin, to appear.
  • 14. Jerome Sacks and Donald Ylvisaker, Statistical designs and integral approximation, Proc. Twelfth Biennial Sem. Canad. Math. Congr. on Time Series and Stochastic Processes; Convexity and Combinatorics (Vancouver, B.C., 1969) Canad. Math. Congr., Montreal, Que., 1970, pp. 115–136. MR 0277069
  • 15. Shoji, I. (1998). Approximation of continuous time stochastic processes by a local linearization method. Math. Comp. 67, 287-298. MR 98e:65207
  • 16. Yingcai Su and Stamatis Cambanis, Sampling designs for estimation of a random process, Stochastic Process. Appl. 46 (1993), no. 1, 47–89. MR 1217687,
  • 17. Talay, D. (1995). Simulation of stochastic differential systems. In Probabilistic Methods in Applied Physics (P. Krée, W. Wedig, eds.) Lecture Notes in Physics 451, Springer, Berlin., 54-96.
  • 18. J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski, Information-based complexity, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1988. With contributions by A. G. Werschulz and T. Boult. MR 958691
  • 19. Wagner, W. and Platen, E. (1978). Approximation of Ito integral equations. Preprint ZIMM, Akad. Wiss. DDR, Berlin.

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Additional Information

Norbert Hofmann
Affiliation: Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstraße 1 1/2, 91054 Erlangen, Germany

Thomas Müller-Gronbach
Affiliation: Mathematisches Institut, Freie Universität Berlin, Arminallee 2–6, 14195 Berlin, Germany

Klaus Ritter
Affiliation: Fakultät für Mathematik und Informatik, Universität Passau, Innstr. 33, 94032 Passau, Germany

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