Dear all,
I'd be very glad if you could help me understand something about the one step analysis. In the attached picture $PP(\lambda)$ means a Poisson Process of rate $\lambda$, and $\tilde X_s = X_{T_1+s}-1$ where $T_1$ is the first arrival time of $X$.
The only thing I can't understand is why can we restrict the integral of the expectation to the interval $[0, u]$, i.e. why is the expected value on $[u, \infty)$ of the independent Poisson Process equal to $0$
Edit I think I know what's going on: If $t > u$, then we are still not having the first arrival, hence the expectation is $0$. Is that the case?