Optimal pointwise approximation of SDE's from inexact information

We study a pointwise approximation of solutions of systems of stochastic differential equations. We assume that an approximation method can use values of the drift and diffusion coefficients which are perturbed by some deterministic noise. Let δ1,δ2≥0 be the precision levels for the drift and diffusion coefficients, respectively. We give a construction of the randomized Euler scheme and we prove that it has the error O(n−min{ϱ,1/2}+δ1+δ2), where n is the number of discretization points and ϱ is the Hölder exponent of the diffusion coefficient. We also investigate lower bounds on the error of an arbitrary algorithm and establish optimality of the defined randomized Euler algorithm. Finally, we report some numerical results.