Continuous-time Markov chains

[Ivo Siekmann]

Often, continuous-time Markov chains with n_S states are defined by the
differential equation

d P(t) / d t = P(t) Q, P(0)=I_{n}, Q \in R^{n x n}

where

q_{ij} >= 0 for i != j and q_{ii} = \sum_{i != j} (- q_{ij}).

The solution of this differential equation is the transition matrix

P(t) = exp( Q t )

of the Markov chain.

However, I have not seen the following definition where p(t) is a
stochastic vector:

d p(t) / d t = p(t) Q, p_0 \in R^n.

where p_0 is a stochastic vector whose positive components sum up to 1
which gives the initial probabilities for the Markov chain being in any
of the n states.

Is there any deeper reason why the matrix form is more common?

Thanks a lot!
Ivo