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

TL;DR

This paper introduces the ETFSP scheme, a method for approximating and bounding exit distributions and occupation measures of continuous-time Markov chains, with proven convergence and applications in biology.

Contribution

The paper presents the ETFSP scheme, providing theoretical guarantees, computable bounds, and demonstrating its effectiveness through biological examples.

Findings

The ETFSP scheme bounds measures from below and converges as truncation expands.

Monotonically decreasing total variation distances ensure improved approximations.

Applications include gene expression timing and species fixation times.

Abstract

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.