### 2003-26

by Müller-Gronbach, Thomas.

**Series:** 2003-26, Preprints

- MSC:
- 60H20 Stochastic integral equations
- 60H10 Stochastic ordinary differential equations

**Abstract:**

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.

**Keywords:**

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