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In general, the analysis of a homogenous or non-homogenous Poisson process $X(t)$ is well known (we can compute waiting time, master equations, etc.). Here denote the rate as $\lambda$ (or $\lambda(t)$ for the non-homogeneous case). Cases where $\lambda$ depends on the current state of $X(t)$ are also standard (e.g. $\lambda(t) = \lambda_{0}X(t)$).

What about the case when the rate $\lambda(t)$ is a random variable, which is dependent on the history of $X(t)$. Has this sort of process been studied? You can imagine many physical applications where this kind of assumption is relevant (e.g. resource consumption). Of course, the system would be non-Markovian, but it seems like it should be simple enough to be tractable.

I guess another way to think of it would be a Poisson process, where the rate is coupled to an SDE, which is dependent on X(t). Not sure if there is a standard name for such systems, and I am just curious if they fall under some other umbrella term (e.g. semi-Markov models, renewal processes, etc.)

user32486
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    This is not necessarily “non-Markovian.” An example is a continuous time birth-death chain with system state $N(t)$ with birth and/or death rates dependent on $N(t)$. So $X(t)$ can be the accumulated arrivals, which could be Poisson lambda (assuming birth rates do not change with $N(t)$) or could depend on $N(t)$ if you prefer (in which case arrivals depend on history in a well defined way). – Michael Jun 06 '25 at 18:39
  • If you want to randomly choose between two different arrival rates every time you transition to a new state, but hold that fixed until the next transition, you could modify the birth-death chain to have “thickness” of size 2, so there are 2 parallel chains (call them high and low) and we choose either high or low state on each transition. A more complex system can indeed be modeled as an embedded Markov chain as you mention. – Michael Jun 06 '25 at 18:46
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    A classical example for this is the exponential Hawkes process, which is Markovian, see my post here (lacking an answer so far but has the source of the statement) https://math.stackexchange.com/questions/3190982/markovian-hawkes-process-elementary-proof – Jfischer Jun 06 '25 at 18:48
  • So what I envision is something like a Poisson process $X(t)$ with rate $\lambda(t)$, and the rate is (say) exponentially decreasing, but the rate of decrease is dependent on $X(t)$.

    I don't think that this can be Markovian, since $\lambda$ depends on the entire history of $X(t)$. Is this really an embedded Markov chain then, since $\lambda(t)$ is not Markovian?

    @Michael Do you have a good resource on embedded Markov chains (I am not an expert by any means).

     – user32486 Jun 07 '25 at 00:59
  • @Jfischer Can you point me to a good introduction to the Hawkes process? I am not sure I immediately understand how the rate is coupled to the state. – user32486 Jun 07 '25 at 01:10
  • I used https://arxiv.org/abs/1507.02822 when I first started learning about them but there might be something more recent. A quick Google search gave https://link.springer.com/book/10.1007/978-3-030-84639-8 but I cannot judge the quality here. – Jfischer Jun 07 '25 at 11:02
  • For simulation I build a tool based on poisson thinning but I never pushed it to become a package https://github.com/jeMATHfischer/TheHawkesPackage/blob/master/Examples%20and%20Use.ipynb – Jfischer Jun 07 '25 at 11:04
  • @user32486 The continuous time Markov chain $X(t)$ with state space $S={0, 1, 2, ...}$ and transition rates $q_{i,i+1}=\lambda/(i+1)$ for all $i \in {0, 1, 2, ...}$ (and $q_{i,j}=0$ if $j\neq i+1$) fits your description. This indeed depends on history, if you want something else you may need to clarify what you mean by "depends on the entire history." For example, maybe you want $\lambda(t) = f(X(t),X(t-1))$ for some positive function $f$ or $\lambda(t) = g(X(t), X(t-0.2), X(t-1.9))$ for some $g$. – Michael Jun 07 '25 at 13:48
  • @Michael Basically what I envision is a process where $\lambda(t)$ solves an SDE related to $X(t)$. For example, if $d\lambda = -\lambda_{0}X(t)\lambda(t) d t + \sigma dW(t)$. Then the entire history of $X$ is needed to predict $\lambda$. – user32486 Jun 08 '25 at 00:47

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