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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!

user565950
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  • It is easier to talk about non-homogeneous Markov chains in discrete time, since the transition probability matrix $(P_{ij})$ is simply replaced by $(P_{ij}(t))$ for $t \in {0, 1, 2, ...}$. For continuous time the transition rates $q_{ij}$ are replaced by $q_{ij}(t)$. This can be intuitively understood by chopping time into small sub-intervals of size $\delta \approx 0$ and thinking of a discrete time chain with $P_{ij}(t_k) \approx q_{ij}(t_k)\delta$, where $t_k$ is the start of the $k$th subinterval. The best way of simulating such things is not obvious. – Michael May 30 '18 at 00:06

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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 $q_{ij}(t+X_i(t))/v_i(t+X_i(t))$.

Michael
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  • Thank you very much! That's a nice way of thinking of it! Although my question was more whether the definition of a non-homogeneous birth-death process can somehow be done in analogy to the definition of a homogeneous birth-death process which defines the transition rates as derivatives of the transition function. – user565950 May 30 '18 at 16:57