A Projected Drift-Randomized Milstein Method for SDEs with Non-differentiable and Super-linear Drift Coefficients

Abstract

We propose a projected drift-randomized Milstein (PRM) method for stochastic differ ential equations with non-differentiable and super-linearly growing drift coefficients. The method extends the randomized Milstein approach beyond the globally Lipschitz setting by incorporating a drift projection into the randomized quadrature approximation. Moreover, unlike existing first-order Milstein-type methods for SDEs with super-linearly growing drift coefficients, the proposed method does not require spatial differentiability of the drift coefficient. Under suitable polynomial Lipschitz and one-sided Lipschitz conditions on the drift, together with standard regularity assumptions on the diffusion, we establish a one-step mean-square stability estimate and derive the required local residual bounds. These esti mates yield first-order strong convergence of the PRM method in the L2-sense. Numerical experiments confirm the theoretical convergence rate and demonstrate the applicability of the method to SDEs with non-differentiable and super-linearly growing drifts.