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Logarithm of a Markov Matrix

It is known that to every Q-matrix corresponds a unique Markov process. Does the converse hold? Specifically,

  1. Can a (discrete state, continuous time) Markov process not be differentiable at the origin, i.e. can $\lim_{t\downarrow0}\frac{1}{t}\mathrm{P}_e[X_t=d]\space\space(*)$ not exist for some states $e,d$?
  2. If $(*)$ exists for all $e,d$, letting $q(e,d)$ denotes this limit, can $q$ fail to exhibit either of the following two properties?

    i) $q(e,d)\geq0$ for all $e,d$,

    ii) $q(e,e)=-\sum_{e\neq d}q(e,d)$

  3. If $(*)$ exists and both $(i)$ and $(ii)$ hold, can the following condition still fail?

    iii) $\sup_{e\in E}|q(e,e)|<\infty$

  4. Does a time-inhomogeneous Markov process have a Q-matrix? If so, is it unique?

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