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2 - Continuous-time Markov chains

Published online by Cambridge University Press:  11 November 2010

Yuri Suhov
Affiliation:
University of Cambridge
Mark Kelbert
Affiliation:
University of Wales, Swansea
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Summary

Q-matrices and transition matrices

Markov processes specialists like to do it with chains.

(From the series ‘How they do it’.)

Definition 2.1.1 A Q-matrix on a finite or countable state space I is a real-valued matrix (qij, i, jI) with:

non-positive diagonal entries qii ≤ 0, iI,

non-negative off-diagonal entries qij ≥ 0, ij, i, jI,

the row zero-sum condition: −qii = ΣjI:jiqij, i.e. Σjqij = 0 for all iI.

For ij, the value qij represents the jump, or transition rate from state i to j. The value −qii = Σj:jiqij is denoted by qi (we will see that it represents the total jump, or exit rate from state i). A Q-matrix will be denoted by Q (a common abuse of notation). As in Chapter 1, we will denote by I the unit matrix.

In a general theory of countable continuous-time Markov chains, the row zerosum condition ΣjI:jiqij = −qii presumes that the series Σj:j≠iqij < ∞. However, a substantial part of the theory can be developed when the equality in this condition is relaxed to the upper bound Σjqij ≤ 0, i.e. qi ≥ Σj:jiqij for all iI. Then a Q-matrix satisfying the row zero-sum condition is called conservative; we will omit this term in the present volume, as we will not consider non-conservative Q-matrices.

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Publisher: Cambridge University Press
Print publication year: 2008

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