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November 2004 Optimal pointwise approximation of SDEs based on Brownian motion at discrete points
Thomas Müller-Gronbach
Ann. Appl. Probab. 14(4): 1605-1642 (November 2004). DOI: 10.1214/105051604000000954


We study pathwise approximation of scalar stochastic differential equations at a single point. We provide the exact rate of convergence of the minimal errors that can be achieved by arbitrary numerical methods that are based (in a measurable way) on a finite number of sequential observations of the driving Brownian motion. The resulting lower error bounds hold in particular for all methods that are implementable on a computer and use a random number generator to simulate the driving Brownian motion at finitely many points. Our analysis shows that approximation at a single point is strongly connected to an integration problem for the driving Brownian motion with a random weight. Exploiting general ideas from estimation of weighted integrals of stochastic processes, we introduce an adaptive scheme, which is easy to implement and performs asymptotically optimally.


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Thomas Müller-Gronbach. "Optimal pointwise approximation of SDEs based on Brownian motion at discrete points." Ann. Appl. Probab. 14 (4) 1605 - 1642, November 2004.


Published: November 2004
First available in Project Euclid: 5 November 2004

zbMATH: 1074.65010
MathSciNet: MR2099646
Digital Object Identifier: 10.1214/105051604000000954

Primary: 65C30
Secondary: 60H10

Keywords: adaptive scheme , asymptotic optimality , pathwise approximation , step-size control , Stochastic differential equations

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.14 • No. 4 • November 2004
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