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Optimal approximation of stochastic differential equations by adaptive step-size control
Author(s):
Norbert
Hofmann;
Thomas
Müller-Gronbach;
Klaus
Ritter.
Journal:
Math. Comp.
69
(2000),
1017-1034.
MSC (1991):
Primary 65U05;
Secondary 60H10
Posted:
May 20, 1999
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Abstract:
We study the pathwise (strong) approximation of scalar stochastic differential equations with respect to the global error in the  -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:
- 1.
- Bouleau, N. and Lépingle, D. (1994). Numerical Methods for Stochastic Processes. Wiley, New York. MR 95h:60090
- 2.
- Cambanis, S. and Hu, Y. (1996). Exact convergence rate of the Euler-Maruyama scheme, with application to sampling design. Stochastics Stochastics Rep. 59, 211-240. MR 97k:60159
- 3.
- Clark, J. M. C. and Cameron, R. J. (1980). The maximum rate of convergence of discrete approximations. In Stochastic Differential Systems (B. Grigelionis, ed.) Lect. Notes Control Inf. Sci. 25. Springer, Berlin, 162-171. MR 82f:60133
- 4.
- Faure, O. (1992). Simulation du mouvement brownien et des diffusions. Thèse ENPC, Paris.
- 5.
- Gaines, J. G. and Lyons, T. J. (1997). Variable step size control in the numerical solution of stochastic differential equations. SIAM J. Appl. Math. 57, 1455-1484. MR 98m:60089
- 6.
- Kloeden, P. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin. MR 94b:60069
- 7.
- Maiorov, V. (1996). About widths of Wiener space in the
 -norm. J. Complexity 12, 47-57. MR 97b:41028 - 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.
- Milstein, G. N. (1995). Numerical Integration of Stochastic Differential Equations. Kluwer, Dordrecht. MR 96e:65003
- 10.
- Müller-Gronbach, T. (1996). Optimal design for approximating the path of a stochastic process. J. Statist. Planning Inf. 49, 371-385. MR 97h:62087
- 11.
- Newton, N. J. (1990). An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times. Stochastics Stochastics Rep. 29, 227-258. MR 91a:60152
- 12.
- Novak, E. and Ritter, K. (1993), Some complexity results for zero finding for univariate functions. J. Complexity 9, 15-40. MR 94c:65179
- 13.
- Ritter, K. (1999). Average Case Analysis of Numerical Problems. Lect. Notes in Math., Springer, Berlin, to appear.
- 14.
- Sacks, J. and Ylvisaker, D. (1970). Statistical design and integral approximation. In Proc. 12th Bienn. Semin. Can. Math. Congr. (R. Pyke, ed.) Can. Math. Soc., Montreal, 115-136. MR 43:2806
- 15.
- Shoji, I. (1998). Approximation of continuous time stochastic processes by a local linearization method. Math. Comp. 67, 287-298. MR 98e:65207
- 16.
- Su, Y. and Cambanis S. (1993). Sampling designs for estimation of random processes. Stochastic Processes Appl. 46, 47-89. MR 95d:62153
- 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.
- Traub, J. F., Wasilkowski. G. W. and Wo\'{z}niakowski, H. (1988). Information-Based Complexity. Academic Press, New York. MR 90f:68085
- 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
Email:
hofmann@mi.uni-erlangen.de
Thomas
Müller-Gronbach
Affiliation:
Mathematisches Institut, Freie Universität Berlin, Arminallee 2--6, 14195 Berlin, Germany
Email:
gronbach@math.fu-berlin.de
Klaus
Ritter
Affiliation:
Fakultät für Mathematik und Informatik, Universität Passau, Innstr. 33, 94032 Passau, Germany
Email:
klaus.ritter@fmi.uni-passau.de
DOI:
10.1090/S0025-5718-99-01177-1
PII:
S 0025-5718(99)01177-1
Keywords:
Stochastic differential equations,
pathwise approximation,
adaption,
step-size control,
asymptotic optimality
Received by editor(s):
August 24, 1998
Posted:
May 20, 1999
Additional Notes:
The first author's work was supported by the DFG:GR 876/9-2.
Copyright of article:
Copyright
2000,
American Mathematical Society
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