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We introduce the exit time finite state projection (ETFSP) scheme, atruncation-based method that yields approximations to the exit distribution andoccupation measure associated with the time of exit from a domain (i.e., thetime of first passage to the complement of the domain) of time-homogeneouscontinuous-time Markov chains. We prove that: (i) the computed approximationsbound the measures from below; (ii) the total variation distances between theapproximations and the measures decrease monotonically as states are added tothe truncation; and (iii) the scheme converges, in the sense that, as thetruncation tends to the entire state space, the total variation distances tendto zero. Furthermore, we give a computable bound on the total variationdistance between the exit distribution and its approximation, and we delineatethe cases in which the bound is sharp. We also revisit the related finite stateprojection scheme and give a comprehensive account of its theoreticalproperties. We demonstrate the use of the ETFSP scheme by applying it to twobiological examples: the computation of the first passage time associated withthe expression of a gene, and the fixation times of competing species subjectto demographic noise.

The exit time finite state projection scheme: bounding exit distributions and occupation measures of continuous-time Markov chains