Independent coupling of two cts time chains.

Independent coupling of two cts time chains.

The trouble is that I care about being in a particular setup called the
"nonexplosive" setup. This is when the continuous time MC can be described
by a Q matrix, whose minimal solution $p_t(., .)$ yields the transition
function for that MC. Otherwise, if I did not care about having a Q matrix
I could just take a cartesian product and prove the markov property for
the product chain. Here is the problem stated formally:
Suppose that $Q_1, Q_2, p_{t,1}, p_{t,2}$ are the parameters for the two
chains on countable state spaces $S_1$ and $S_2$. I must check that if I
obtain $Q$ the Q-matrix by setting it:
$q((x_1, x_2), (y_1, y_2))=\delta_{x_1y_1}q_2(x_2,
y_2)+\delta_{x_2y_2}q_1(x_1, y_1)$
then I have to see that the solution to the Kolmogorov backwards eqn
corresponding to this $Q$ is $p_t((x_1, x_2), (y_1, y_2))=p_{t, 1}(x_1,
y_1)p_{t, 2}(x_2, y_2)$
I have tried many things to show that the suggested solution is the
minimal one. On the one hand, it does solve the required initial value
problem. (KBE for $Q$) In other words, that proves $\leq$ above in the
last display. On the other hand, to see that it's smallest, what I've
tried include things like summing on $y_2$ on the LHS and seeing if it
satisfies the KBE for $Q_1$. I have not been able to check this. I have
also tried to use more probabilistic methods like thinking about the
discrete embedded chain.