Randomized derivative-free Milstein algorithm for efficient approximation of solutions of SDEs under noisy information
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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 A̅^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 A̅^df-RM_n is the optimal one. Finally, in order to test the algorithm A̅^df-RM_n in practice, we report performed numerical experiments.
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