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Best Rates of Convergence for Strong Approximation of SDE's at a Single Point

by Müller-Gronbach, Thomas.

Series: 2003-26, Preprints

60H20 Stochastic integral equations
60H10 Stochastic ordinary differential equations

We study pathwise approximation of scalar sde's 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, e.g., via C-codes, and use a standard normal 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 optimal.

stochastic differential equations, pathwise approximation, adaptive scheme, step-size control, asymptotic optimality

This paper was published in:
Annals of Applied Probability, Vol. 14, No. 4, 2004