I think this is basically the inverse question of [Matrices whose exponential is stochastic](https://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.



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


