On explicit order 1.5 approximations with varying coefficients: the case of super-linear diffusion coefficients

TL;DR

This paper affirms the possibility of constructing explicit order 1.5 numerical schemes for SDEs with superlinear coefficients, validating a conjecture and exploring cases with Hölder continuous derivatives.

Contribution

It provides a constructive approach to high-order explicit schemes for SDEs with superlinear coefficients, confirming a conjecture and extending to Hölder continuous derivatives.

Findings

Constructed explicit order 1.5 schemes for superlinear SDEs.

Validated the methodology proposed in the conjecture.

Extended analysis to cases with Hölder continuous derivatives.

Abstract

A conjecture appears in \cite{milsteinscheme}, in the form of a remark, where it is stated that it is possible to construct, in a specified way, any high order explicit numerical schemes to approximate the solutions of SDEs with superlinear coefficients. We answer this conjecture affirmatively for the case of order 1.5 approximations and show that the suggested methodology works. Moreover, we explore the case of having H\"{o}lder continuous derivatives for the diffusion coefficients.