  We introduce the exit time finite state projection (ETFSP) scheme, a
truncation-based method that yields approximations to the exit distribution and
occupation measure associated with the time of exit from a domain (i.e., the
time of first passage to the complement of the domain) of time-homogeneous
continuous-time Markov chains. We prove that: (i) the computed approximations
bound the measures from below; (ii) the total variation distances between the
approximations and the measures decrease monotonically as states are added to
the truncation; and (iii) the scheme converges, in the sense that, as the
truncation tends to the entire state space, the total variation distances tend
to zero. Furthermore, we give a computable bound on the total variation
distance between the exit distribution and its approximation, and we delineate
the cases in which the bound is sharp. We also revisit the related finite state
projection scheme and give a comprehensive account of its theoretical
properties. We demonstrate the use of the ETFSP scheme by applying it to two
biological examples: the computation of the first passage time associated with
the expression of a gene, and the fixation times of competing species subject
to demographic noise.
