Randomized derivative-free Milstein algorithm for efficient approximation of solutions of SDEs under noisy information

We deal with pointwise approximation of solutions of scalar stochastic differential equations in the presence of informational noise about underlying drift and diffusion coefficients. We define a randomized derivative-free version of Milstein algorithm $\mathcal{\bar A}^{df-RM}_n$ and investigate its error. We also study lower bounds on the error of an arbitrary algorithm. It turns out that in some case the scheme $\mathcal{\bar A}^{df-RM}_n$ is the optimal one. Finally, in order to test the algorithm $\mathcal{\bar A}^{df-RM}_n$ in practice, we report performed numerical experiments.