"Forcing" equal probabilities in the matrix exponential of a Markov intensity matrix

Question

I have an upper-right triangular transition intensity matrix $Q$ for a 7-state Markov model (with states $X_1,X_2,...,X_7$), from which I numerically calculate the matrix exponential to derive a transition probability matrix $P$.

A number of the transitions into one of the states have equal rates in the intensity matrix, say: $$q_{1,6} = q_{2,6} = q_{3,6}$$ The matrix is otherwise populated such that, after taking the matrix exponential, the derived probabilities of transitioning into $X_6$ from states $X_1,X_2,X_3$ are then different, i.e.: $$p_{1,6} ≠ p_{2,6} ≠ p_{3,6}$$

This is, of course, an expected consequence of taking the matrix exponential; however, for the purposes of a simulation I am trying to run, it is undesirable as it results in different proportions in $X_6$ in two different "arms" of the simulation based on the same Markov structure, but with different initial distributions across $X_1,X_2,X_3$.

Putting concerns around competing risks aside, is there a method by which the intensity matrix can be adjusted (or indeed any other method) to ensure that the derived probabilities of transitioning into a specific state are the same where the underlying rates are the same? (i.e. to ensure that $p_{1,6} = p_{2,6} = p_{3,6}$ when $q_{1,6} = q_{2,6} = q_{3,6}$).

Answer 1

There's no reason to expect these equalities to be preserved.

• The intensity matrix answers the question of "If we are in state $1$, with what frequency will state $2$ be the next state?".
• The transition matrix answers the question of "If we are in state $1$ now, what is the probability of being in state $2$ some unit of time later?".

The key difference here is that in the second case we can take any route from $1$ to $2$. (i.e. we can go $1 \implies 3 \implies 2$ or any other path).

So the transition probabilities $p_{16}$, $p_{26}$ and $p_{36}$ will be different as states $1$, $2$ and $3$ have different probabilities of taking the various paths.

You should however be able to observe that after taking the matrix exponential that the equilibrium probabilities of each state are preserved which seems to be what you are really interested in.