A tamed-adaptive Milstein scheme for stochastic differential equations with low regularity coefficients

TL;DR

This paper introduces a tamed-adaptive Milstein numerical scheme for stochastic differential equations with low regularity coefficients, achieving convergence with a rate depending on the local Hölder continuity of the derivatives.

Contribution

The paper develops a novel tamed-adaptive Milstein scheme that converges for SDEs with coefficients having Hölder continuous derivatives, extending numerical methods to less regular cases.

Findings

Converges in L2-norm with rate (1+α)/2

Applicable over finite and infinite time intervals

Works under certain growth conditions on coefficients

Abstract

We propose a tamed-adaptive Milstein scheme for stochastic differential equations in which the first-order derivatives of the coefficients are locally H\"older continuous of order $\alpha$. We show that the scheme converges in the $L_2$-norm with a rate of $(1+\alpha)/2$ over both finite intervals $[0, T]$ and the infinite interval $(0, +\infty)$, under certain growth conditions on the coefficients.