  Motivated by nucleation and molecular aggregation in physical, chemical and
biological settings, we present a thorough analysis of the general problem of
stochastic self-assembly of a fixed number of identical particles in a finite
volume. We derive the Backward Kolmogorov equation (BKE) for the cluster
probability distribution. From the BKE we study the distribution of times it
takes for a single maximal cluster to be completed, starting from any initial
particle configuration. In the limits of slow and fast self-assembly, we
develop analytical approaches to calculate the mean cluster formation time and
to estimate the first assembly time distribution. We find, both analytically
and numerically, that faster detachment can lead to a shorter mean time to
first completion of a maximum-sized cluster. This unexpected effect arises from
a redistribution of trajectory weights such that upon increasing the detachment
rate, paths that take a shorter time to complete a cluster become more likely.
