I have a stochastic point process with event times $\{x_1, x_2, ...\} $ and I want to compute the expected number of events $n(T)$ over the interval $[0,T]$. The point process is generated as follows:

  1. The first event is distributed according to density $f(x)$ over $x \in (0,\infty)$.
  2. Conditioned on event $x_n$:

    (i) With probability $\alpha$, $x_{n+1}$ is distributed according to the truncated density $f(x > x_n)$. The truncared distribution is defined as $\frac{f(x)}{1- F(x_n)}$ for $x \in (x_n, \infty)$.

    (ii) With probability $1-\alpha$, $x_{n+1} = x_n + d_n$ where $d_n$ is distributed according to $f(x)$, which is similar to a renewal process.

How can we compute the expected number of points over the interval $[0,T]$? I tried to write the following equation:

$n(T) = \alpha \int\limits_{x_1 = 0}^T f(x_1) (n(T-x_1)+1) dx_1$

$ + (1-\alpha) \int\limits_{x_1 = 0}^T \hspace{2mm} \int\limits_{x_2 = x_1}^T f(x_1) \frac{f(x_2)}{1-F(x_1)} (n(T-x_2)+2) dx_1 dx_2$

Is it possible to solve this equation or is there a better way to solve this problem? Also, if it helps, you may also use the rate definition of point processes, where each event arrives at a rate $\lambda (x |x_{n})$. Here, $x_n$ is the latest event.


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