A modified Milstein scheme for approximation of stochastic delay differential equations with constant time lag

We introduce a modified Milstein scheme for pathwise approximation of scalar stochastic delay differential equations with constant time lag on a fixed finite time interval. Our algorithm is based on equidistant evaluation of the driving Brownian motion and is simply obtained by replacing iterated Itô-integrals by products of appropriate Brownian increments in the definition of the Milstein scheme. We prove that the piecewise linear interpolation of the modified Milstein scheme is asymptotically optimal with respect to the mean square L2L2-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 modified Milstein scheme even for a very small number of discretization points. Finally, we provide a generalization of our approach to the case of a system of SDDEs with an arbitrary finite number of constant delays. We conjecture that the above optimality results carry over to the generalized scheme.