Particle splitting methods are considered for the estimation of rare events.
The probability of interest is that a Markov process first enters a set $B$
before another set $A$, and it is assumed that this probability satisfies a
large deviation scaling. A notion of subsolution is defined for the related
calculus of variations problem, and two main results are proved under mild
conditions. The first is that the number of particles generated by the
algorithm grows subexponentially if and only if a certain scalar multiple of
the importance function is a subsolution. The second is that, under the same
condition, the variance of the algorithm is characterized (asymptotically) in
terms of the subsolution. The design of asymptotically optimal schemes is
discussed, and numerical examples are presented.