2005/03/24 by Thomas Muller-Gronbach · 1 citation
Mathematics · #math.PR #msc:65C30 #msc:60H10.
paper · pdf · doi:10.1214/105051604000000954
published as Annals of Applied Probability 2004, Vol. 14, No. 4, 1605-1642 · Published at http://dx.doi.org/10.1214/105051604000000954 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
arxiv created 2005/03/24 · arxiv updated 2009/12/01
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.