Optimal pointwise approximation of SDEs based on brownian motion at discrete points

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dc.creatorMuller-Gronbach, Thomas
dc.date2005-03-24
dc.date.accessioned2026-07-07T05:18:20Z
dc.date.available2026-07-07T05:18:20Z
dc.descriptionWe 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.
dc.descriptionPublished 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)
dc.identifierhttps://arxiv.org/abs/math/0503531
dc.identifierhttp://arxiv.org/abs/math/0503531
dc.identifierAnnals of Applied Probability 2004, Vol. 14, No. 4, 1605-1642
dc.identifierdoi:10.1214/105051604000000954
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/74628
dc.subjectProbability
dc.subject65C30 (Primary) 60H10. (Secondary)
dc.titleOptimal pointwise approximation of SDEs based on brownian motion at discrete points
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