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Let $(X_n)_{n\in\mathbb{N}_0}$ be a time-homogeneus Markov chain, evoloving on a space $X$, with transition law $(x,A)\mapsto P(x,A)$. Further, let $\tau_n: \Omega\to \left[0,\infty\right)$, $n\in\mathbb{N}_0$, be a strictly increasing sequence of random variables with $\tau_0=0$ and $\lim_{n\to\infty} \tau_n=\infty$ (a.s.), whose increments $\Delta \tau_{n+1}:=\tau_{n+1} - \tau_n$ are mutually independent and have the conditional distribution of the form

$$\mathbb{P}(\Delta \tau_{n+1} \leq t\, | \, X_n=x)=1-\exp\left(-\int_0^t \lambda(x,s)\,ds\right).$$

Consider the process $(N_t)_{t\geq 0}$ defined by $$N_t=\sum_{n=0}^{\infty} \boldsymbol{1}_{\{\tau_n\leq t\}},$$ that is $\{N_t=n\}=\{\tau_n\leq t < \tau_{n+1}\}$. Clearly, if $\lambda$ is constant, then $(N_t)_{t\geq 0}$ is just a Poisson counting process.

Is anything known about $(N_t)_{t\geq 0}$ in the general case? Is there any book considering such processes? Specifically, I am interested in the following issues:

  1. Do the increments of $(N_t)_{t\geq 0}$ are conditionally independent given $X_0$?

  2. Is it possible to compute $\mathbb{P}(N_t=n\, | \, X_0=x)$ in terms of $\lambda$ and $P$?

  3. Can we determine the conditional distribution of $(\Delta \tau_1,\ldots,\Delta \tau_n)$ given $\{N_t=n, X_0=x\}$ (for a Poisson process it is the same as the joint distribution of the order statistics of $n$ i.i.d. random variables with the uniform distribution on $(0,t)$)?

I would be grateful for any information.

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