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

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 a continuous-time Markov chain. We prove that: (i) the computed approximations bound the desired 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. To establish the theoretical properties of the ETFSP scheme, we revisit the related finite state projection scheme taking a probabilistic viewpoint and proving the latter’s convergence. 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.