A Randomized Milstein Scheme for SDEs with Superlinear Drift Coefficient

TL;DR

This paper introduces a novel randomized-tamed Milstein scheme for SDEs with superlinear drift and limited regularity, achieving optimal convergence rates by combining taming and drift randomization techniques.

Contribution

It proposes a new numerical scheme that effectively handles superlinear growth and low regularity in SDEs, improving convergence performance.

Findings

Achieves strong $ ext{L}^p$-convergence rate of order one.

Successfully manages superlinear drift with limited temporal regularity.

Combines taming and randomization for enhanced numerical stability.

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 $\beta$-H\"older continuity with $\beta \in (0,1]$. 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 $\mathscr{L}^p$-convergence rate of order one.