  This paper examines convergence and stability of the two classes of
theta-Milstein schemes for stochastic differential equations (SDEs) with
non-global Lipschitz continuous coefficients: the split-step theta-Milstein
(SSTM) scheme and the stochastic theta-Milstein (STM) scheme. For
\theta\in[1/2,1], this paper concludes that the two classes of theta-Milstein
schemes converge strongly to the exact solution with the order 1. For \theta
\in [0,1/2], under the additional linear growth condition for the drift
coefficient, these two classes of the theta-Milstein schemes are also strongly
convergent with the standard order. This paper also investigates exponential
mean-square stability of these two classes of the theta-Milstein schemes. For
\theta\in(1/2, 1], these two theta-Milstein schemes can share the exponential
mean-square stability of the exact solution. For \theta\in[0, 1/2], similar to
the convergence, under the additional linear growth condition, these two
theta-Milstein schemes can also reproduce the exponential mean-square stability
of the exact solution.
