## Journal of Applied Probability

- J. Appl. Probab.
- Volume 48, Number 4 (2011), 911-924.

### Summary statistics for endpoint-conditioned continuous-time Markov chains

Asger Hobolth and Jens Ledet Jensen

#### Abstract

Continuous-time Markov chains are a widely used modelling tool. Applications include DNA sequence evolution, ion channel gating behaviour, and mathematical finance. We consider the problem of calculating properties of summary statistics (e.g. mean time spent in a state, mean number of jumps between two states, and the distribution of the total number of jumps) for discretely observed continuous-time Markov chains. Three alternative methods for calculating properties of summary statistics are described and the pros and cons of the methods are discussed. The methods are based on (i) an eigenvalue decomposition of the rate matrix, (ii) the uniformization method, and (iii) integrals of matrix exponentials. In particular, we develop a framework that allows for analyses of rather general summary statistics using the uniformization method.

#### Article information

**Source**

J. Appl. Probab., Volume 48, Number 4 (2011), 911-924.

**Dates**

First available in Project Euclid: 16 December 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1324046009

**Digital Object Identifier**

doi:10.1239/jap/1324046009

**Mathematical Reviews number (MathSciNet)**

MR2896658

**Zentralblatt MATH identifier**

1231.60071

**Subjects**

Primary: 60-08: Computational methods (not classified at a more specific level) [See also 65C50]

Secondary: 60J22: Computational methods in Markov chains [See also 65C40] 60J25: Continuous-time Markov processes on general state spaces 60J27: Continuous-time Markov processes on discrete state spaces

**Keywords**

Continuous-time Markov chain dwelling time EM algorithm transition number uniformization

#### Citation

Hobolth, Asger; Jensen, Jens Ledet. Summary statistics for endpoint-conditioned continuous-time Markov chains. J. Appl. Probab. 48 (2011), no. 4, 911--924. doi:10.1239/jap/1324046009. https://projecteuclid.org/euclid.jap/1324046009