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

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 $α$. We show that the scheme converges in the $L_2$-norm with a rate of $(1+α)/2$ over both finite intervals $[0, T]$ and the infinite interval $(0, +\infty)$, under certain growth conditions on the coefficients.