A Randomized Milstein Scheme for SDEs with Superlinear Drift Coefficient

Abstract

This work presents a randomized-tamed Milstein scheme for stochastic differential equations whose drift coefficient exhibits superlinear growth in the state variable and limited temporal regularity, quantified by ${\hat{I}}^2$-Hölder continuity with ${\hat{I}}^2\in (0,1]$. The scheme combines a taming mechanism to control the superlinear state dependence with a drift randomization strategy designed to address the challenges posed by low temporal regularity. Under suitable assumptions on temporal smoothness, the scheme achieves an optimal strong ${\mathcal{L}}^p$-convergence rate of order one.