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I think this is basically the inverse question of http://mathoverflow.net/questions/33230/matrices-whose-exponential-is-stochastic.

i.e. what are sufficient conditions on the matrix representation of an evolution operator of a (finite) discrete Markov chain for it to be embeddable in a continuous Markov chain?

I've found some old paper that may answer this (something about embeddability criteria) but I can't access it as it published in a closed-access journal.

I hope this is a sane question.

flag	asked Aug 31 2010 at 16:28 mathtick 101●6

Furthermore, as a modeling bonus question. Is there, in general, a "right" thing to do when you are given one Markov chain and need to pick a transition matrix for a new Markov chain with shorter time intervals? – mathtick Aug 31 2010 at 16:31

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arxiv.org/abs/1001.1693 – Steve Huntsman Aug 31 2010 at 17:01

mathtick · Accepted Answer · 2010-08-31 19:08:30Z UTC

Steve Hunstman's link above is good:

See the part leading up to Theorem 9 for something relevant to applications:

The main application of the following theorem may be to establish that certain Markov matrices arising in applications are not embeddable, and hence either that the entries are not numerically accurate or that the underlying process is not autonomous. The theorem is a quantitative strengthening of Lemma 8. It is of limited value except when n is fairly small, but this is often the case in applications.

Also the part on regularization for best compromises when matrices are not embeddable.

(Stochastic) matrix for which a stochastic matrix logarithm exists?

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