The Exit Time Finite State Projection Scheme: Bounding Exit Distributions and Occupation Measures of Continuous-Time Markov Chains

Abstract

We introduce the exit time finite state projection (ETFSP) scheme, a truncationbased 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.

The pathway elaboration method for mean first passage time estimation in large continuous-time Markov chains with applications to nucleic acid kinetics

“…Efficiently quantifying the error of MFPT estimates in truncated CTMCs for exponential and non-exponential decay processes is beyond the scope of this paper. It might be possible to use some other existing work (Kuntz et al, 2019;Backenköhler, Bortolussi and Wolf, 2019).…”

“…CTMC systems may be treated by methods that truncate the state space to just a subset of "most relevant" states, so long as those states can be identified and they are few enough that matrix methods can compute the MFPT or other properties of interest (Munsky and Khammash, 2006;Kuntz et al, 2019). In such truncation-based methods, after the initial cost of enumerating states, the truncated CTMC can be reused to compute MFPTs for mildy perturbed parameters with accuracy relying on the truncated state space still containing the most relevant states.…”

Stationary distributions of continuous-time Markov chains: a review of theory and truncation-based approximations

“…is known as the explosion time. It is [83,Lemma 2.1] the instant by which the chain has left every finite subset of the state space (or infinity should this event never occur). In the case of an SRN (2.1), an explosion occurs if and only if the count of at least one species diverges to infinity in a finite amount of time.…”

Bounding Mean First Passage Times in Population Continuous-Time Markov Chains

“…Considerable effort has been directed at the analysis of first passage time distributions in PCTMCs. Most works can either focus on an explicit state-space analysis [5,39,33,32] or employ approximation techniques for which, in general, no error bounds can be given [45,25,11]. For some model classes such as kinetic proofreading, analytic solutions are possible [39,6,29].…”

“…In Ref. [33] the authors propose a finite state-space projection scheme to bound first passage time distributions Hayden et al [25] use moment closure approximations and Chebychev's inequality to gain an understanding of first passage time dynamics. Schnoerr et al [45] also employ a moment closure approximation and further approximate threshold functions to derive an approximate first passage time distribution.…”