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Consider a continuous time Markov chain $(X_t)_{t \geq 0}$ on some state space $S$ with transition matrix (P-matrix) $p_t(x,y)$, the probability density of jumping from $x$ to $y$ in time $t$.

The associated Q-matrix is defined as

$$q(x,y) = \frac{d}{d t}p_t(x,y)\vert_{t=0}$$

and offers an infinitesimal description of the chain.

What are advantages and disadvantages of investigating the Markov with either the Q- or the P-matrix?

Are the two descriptions any different in the case of $S$ being finite or do the methods only differ for $S$ being countably infinite?

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Those are two equivalent descriptions of the same semi-group. You mentioned how to deduce $q$ from $(p_t)$, note that, conversely, $p_t=\mathrm e^{tq}$ for every $t\geqslant0$.

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  • $\begingroup$ Do there have to be assumptions on $Q$ for the exponential series to converge? (which would imply that there is no one-to-one correspondence) Is one description favorable over the other from a practical point of view when working with Markov chains? $\endgroup$ – madison54 Jul 20 '14 at 10:06
  • $\begingroup$ The exponential series always converges, for the same reason that in the scalar case, plus the submultiplicativity of the norm. "More or less favorable": this all depends on what one wants to do with these. $\endgroup$ – Did Jul 20 '14 at 10:11

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