by Hofmann, Norbert, Müller-Gronbach, Thomas.

**Series:** 2004-22, Preprints

- MSC:
- 60H10 Stochastic ordinary differential equations

**Abstract:**

We introduce a new scheme for pathwise approximation of scalar stochastic delay differential equations with constant time lag. Our algorithm is based on equidistant evaluation of the driving Brownian motion and is simply obtained by replacing iterated Ito-integrals by products of

appropriate Brownian increments in the definition of the Milstein scheme. We prove that the piecewise linear interpolation of the new scheme is asymptotically optimal with respect to the mean square $L_2$-error within the class of all pathwise approximations that use observations of the driving Brownian motion at equidistant points. Moreover,

for a large class of equations our scheme is also asymptotically optimal for mean square approximation of the solution at the final time point. Our asymptotic optimality results are complemented by a comparison with the Euler scheme based on exact error formulas for a linear test

equation. This comparison demonstrates the superiority of the new method even for a very small number of discretization points.

**Keywords:**

stochastic delay differential equations, pathwise approximation, asymptotic optimality, minimal errors, exact error formulas