How to model a continuous time process that is, at every instant, a positive constant (say, $δ > 0$) but at random arrival times jumps to zero? Let’s denote with ${N(t), t ≥ 0}$ the number of jumps from $δ$ to 0 in [0, t], and model it as a Poisson counting process with intensity parameter $η > 0$. That is, $dN(t)$ is $0$ or $1$.
My attempt is: $$dD(t)=\delta dt-\delta dN(t),$$ where $D(t)$ is the cumulative of the process.
Does this imply that, in a small time interval $[t,t+dt]$, with probability $\eta dt$, the random variable is $0$, but with probability $1-\eta dt$, it is $\delta$? Put together, then, the expectation is $$\eta dt (0)+(1-\eta dt)\delta=(1-\eta dt)\delta.$$
If I take the expectation of the first equation directly: $$E[dD(t)]=\delta dt - \delta \eta dt=\delta(1-\eta)dt.$$
This is, however, inconsistent with the second equation.
So my question is (1) is my attempt correct? (2) if it is, in a small time interval, what is the probability it is $\delta$ vs. $0$, and (3) what is its expectation?