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

Abstract

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