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.
@article{1099674072,
author = {M\"uller-Gronbach, Thomas},
title = {Optimal pointwise approximation of SDEs based on Brownian motion at discrete points},
journal = {Ann. Appl. Probab.},
volume = {14},
number = {1},
year = {2004},
pages = { 1605-1642},
language = {en},
url = {http://dml.mathdoc.fr/item/1099674072}
}
Müller-Gronbach, Thomas. Optimal pointwise approximation of SDEs based on Brownian motion at discrete points. Ann. Appl. Probab., Tome 14 (2004) no. 1, pp. 1605-1642. http://gdmltest.u-ga.fr/item/1099674072/