On Milstein approximations with varying coefficients: the case of super-linear diffusion coefficients

TL;DR

This paper introduces a new class of explicit Milstein schemes for SDEs with superlinear coefficients, proving their convergence with optimal rate under mild conditions, advancing numerical methods for complex stochastic systems.

Contribution

The paper proposes a novel explicit Milstein scheme tailored for SDEs with superlinear coefficients, demonstrating its convergence and optimal rate under mild assumptions.

Findings

Schemes converge in p to SDE solutions

Convergence holds under very mild conditions

Achieves optimal convergence rate

Abstract

A new class of explicit Milstein schemes, which approximate stochastic differential equations (SDEs) with superlinearly growing drift and diffusion coefficients, is proposed in this article. It is shown, under very mild conditions, that these explicit schemes converge in $\mathcal L^p$ to the solution of the corresponding SDEs with optimal rate.