  Milestoning is a computational procedure that reduces the dynamics of complex
systems to memoryless jumps between intermediates, or milestones, and only
retains some information about the probability of these jumps and the time lags
between them. Here we analyze a variant of this procedure, termed optimal
milestoning, which relies on a specific choice of milestones to capture exactly
some kinetic features of the original dynamical system. In particular, we prove
that optimal milestoning permits the exact calculation of the mean first
passage times (MFPT) between any two milestones. In so doing, we also analyze
another variant of the method, called exact milestoning, which also permits the
exact calculation of certain MFPTs, but at the price of retaining more
information about the original system's dynamics. Finally, we discuss
importance sampling strategies based on optimal and exact milestoning that can
be used to bypass the simulation of the original system when estimating the
statistical quantities used in these methods.
