I know that the usual definition of a birth-death processes found in books uses a homogeneous Markov processes, defines a transition function and uses the derived q-matrix to define the birth and death rates.
Clearly a thing as a non-homogeneous birth-death processes does exist which means, that the rates are time-dependant but I wonder how it is defined as we can't find a transition function in the first place?
Most books that I tried to find the answer in only defined the homogeneous case and suddenly started talking about non-homogeneous ones while I can't see a rigorous transition between the cases.
Am I missing something obvious? Thank you very much!
After some thought, it seems the following is a good way to simulate:
For homogeneous CTMCs on a finite or countably infinite state space $S$ we have transition rates $(q_{ij})$ and define $v_i = \sum_{j\neq i} q_{ij}$ as the sum outgoing rate of each state $i \in S$. We can simulate by: Given we are in state $i$, generate an exponentially distributed random variable $X_i$ with rate $v_i$. After time $X_i$, independently transition to a next-state $j$ with probability $q_{ij}/v_i$.
So for inhomogeneous CTMCs: Define $v_i(u) = \sum_{j\neq i} q_{ij}(u)$ for all time $u\geq 0$. This is the instantaneous sum rate out of state $i$. Given we enter state $i$ at time $t$, generate a random variable $X_i(t)$ with distribution $P[X_i(t)>T] = e^{-\lambda_i(t,T)}$ for all $T>0$, where $$ \lambda_i(t,T) = \int_t^{t+T} v_i(u)du$$ After $X_i(t)$ expires (at time $t+X_i(t)$), independently transition to a next-state $j$ with probability $q_{ij}(t+X_i(t))/v_i(t+X_i(t))$.