Error of randomized Milstein scheme for scalar SDEs with noisy information about coefficients and Wiener process

We investigate the strong approximation of scalar stochastic differential equations when the available standard information about the drift coefficient, the diffusion coefficient, the derivative of the diffusion coefficient, and the observed Wiener path is corrupted by noise. The precision of the drift, diffusion-information, and Wiener-path observations is described by three nonnegative parameters $\delta_1,\delta_2,\delta_3$, where $\delta_2$ controls both the noisy diffusion coefficient and the separate noisy derivative oracle required in the Milstein correction. We analyze a randomized Milstein scheme based only on this noisy information and prove, for $r\geq 2$, that its $L^r$-error is bounded by $C(n^{-\min\{\gamma_1+1/2,\gamma_2\}}+\delta_1+\delta_2+\delta_3)$, where $n$ is the number of time steps and $\gamma_1,\gamma_2$ are the temporal H\"older exponents of the coefficients. We also prove a matching minimax lower bound in the randomized standard-information model considered in the paper. In particular, the Wiener-path contribution proportional to $\delta_3$ is unavoidable, and the noisy randomized Milstein scheme is minimax order-optimal.