Optimal approximation of stochastic differential equations by adaptive step-size control

Hofmann, Norbert; Müller-Gronbach, Thomas; Ritter, Klaus

doi:10.1090/S0025-5718-99-01177-1

Optimal approximation of stochastic differential equations by adaptive step-size control
HTML articles powered by AMS MathViewer

by Norbert Hofmann, Thomas Müller-Gronbach and Klaus Ritter;

Math. Comp. 69 (2000), 1017-1034

DOI: https://doi.org/10.1090/S0025-5718-99-01177-1

Published electronically: May 20, 1999

PDF | Request permission

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.

References

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
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, DOI 10.1080/17442509608834090
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) Lect. Notes Control Inf. Sci., vol. 25, Springer, Berlin-New York, 1980, pp. 162–171. MR 609181
Faure, O. (1992). Simulation du mouvement brownien et des diffusions. Thèse ENPC, Paris.
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, DOI 10.1137/S0036139995286515
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, DOI 10.1007/978-3-662-12616-5
V. E. Maiorov, About widths of Wiener space in the $L_q$-norm, J. Complexity 12 (1996), no. 1, 47–57. MR 1386593, DOI 10.1006/jcom.1996.0006
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.
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, DOI 10.1007/978-94-015-8455-5
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, DOI 10.1016/0378-3758(95)00017-8
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, DOI 10.1080/17442509008833616
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, DOI 10.1006/jcom.1993.1003
Ritter, K. (1999). Average Case Analysis of Numerical Problems. Lect. Notes in Math., Springer, Berlin, to appear.
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, QC, 1970, pp. 115–136. MR 277069
Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
Yingcai Su and Stamatis Cambanis, Sampling designs for estimation of a random process, Stochastic Process. Appl. 46 (1993), no. 1, 47–89. MR 1217687, DOI 10.1016/0304-4149(93)90085-I
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.
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
Wagner, W. and Platen, E. (1978). Approximation of Ito integral equations. Preprint ZIMM, Akad. Wiss. DDR, Berlin.