  We introduce an explicit adaptive Milstein method for stochastic differential
equations (SDEs) with no commutativity condition. The drift and diffusion are
separately locally Lipschitz and together satisfy a monotone condition. This
method relies on a class of path-bounded time-stepping strategies which work by
reducing the stepsize as solutions approach the boundary of a sphere, invoking
a backstop method in the event that the timestep becomes too small. We prove
that such schemes are strongly $L_2$ convergent of order one. This order is
inherited by an explicit adaptive Euler-Maruyama scheme in the additive noise
case. Moreover we show that the probability of using the backstop method at any
step can be made arbitrarily small. We compare our method to other fixed-step
Milstein variants on a range of test problems.
