  A class of implicit Milstein type methods is introduced and analyzed in the
present article for stochastic differential equations (SDEs) with non-globally
Lipschitz drift and diffusion coefficients. By incorporating a pair of method
parameters $\theta, \eta \in [0, 1]$ into both the drift and diffusion parts,
the new schemes are indeed a kind of drift-diffusion double implicit methods.
Within a general framework, we offer upper mean-square error bounds for the
proposed schemes, based on certain error terms only getting involved with the
exact solution processes. Such error bounds help us to easily analyze
mean-square convergence rates of the schemes, without relying on a priori
high-order moment estimates of numerical approximations. Putting further
globally polynomial growth condition, we successfully recover the expected
mean-square convergence rate of order one for the considered schemes with
$\theta \in [\tfrac12, 1], \eta \in [0, 1]$. Also, some of the proposed schemes
are applied to solve three SDE models evolving in the positive domain $(0,
\infty)$. More specifically, the particular drift-diffusion implicit Milstein
method ($ \theta = \eta = 1 $) is utilized to approximate the Heston
$\tfrac32$-volatility model and the stochastic Lotka-Volterra competition
model. The semi-implicit Milstein method ($\theta =1, \eta = 0$) is used to
solve the Ait-Sahalia interest rate model. Thanks to the previously obtained
error bounds, we reveal the optimal mean-square convergence rate of the
positivity preserving schemes under more relaxed conditions, compared with
existing relevant results in the literature. Numerical examples are also
reported to confirm the previous findings.
