Background
I am modeling a two-variable, birth-death process with variables $X(t)>0$ and $Y(t)>0$.
I am assuming that when $X(t)$ gives birth, a $Y(t)$ death occurs. Specifically, one $X(t)$ birth event induces the following system state changes:
$X(t) \rightarrow X(t) + 1$
$Y(t) \rightarrow Y(t) - 1$.
Finally, I assume this single event (and so its two state changes) occurs at a rate of $V(t)>0$.
Question
Is this model equivalent to one in which one $X(t)$ birth induces only $X(t) \rightarrow X(t) + 1$ at a rate of $V(t)$, while one $Y(t)$ death induces $Y(t) \rightarrow Y(t) - 1$ at a rate of $V(t)$?
In other words, is the model in which one event leads to two state changes at a rate of $V(t)$ equivalent to a model where the two state changes correspond to separate events that occur at the same rate $V(t)$?
These two models are not equivalent, chiefly because the births and deaths are uncorrelated in the separate events case and not in the double event case.
Simply note that, if the total initial population of either system is $z$, then for the double-event case, $\mathbb{P}(X(t) + Y(t) \neq z\ |\ t = T) = 0$ for all times $T$. In the separate events case, it is perfectly possible for the sum of both populations to be unequal to $z$ even though the births and deaths cancel out on average.