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

TL;DR

This paper introduces a randomized derivative-free Milstein algorithm for approximating solutions of scalar SDEs with noisy data, demonstrating its optimality in certain cases and validating it through numerical experiments.

Contribution

The paper proposes a new randomized derivative-free Milstein algorithm for SDEs with noisy information and establishes its optimality in some scenarios.

Findings

The algorithm achieves optimal error bounds in certain cases.

Numerical experiments confirm the practical effectiveness of the proposed method.

Lower bounds on approximation error are derived for comparison.

Abstract

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.