Let (left(N^{i}, i=1,2 ight)) be two independent Poisson processes. Prove that (N=N^{1}+N^{2}) is a Poisson process. Compute

Let \(\left(N^{i}, i=1,2\right)\) be two independent Poisson processes. Prove that \(N=N^{1}+N^{2}\) is a Poisson process. Compute the compensator of \(N\). Let \(\tau^{i}=\inf \left\{t: N_{t}^{i}=1\right\}\) and \(\tau=\inf \left\{t: N_{t}=1\right\}\). Compute \(\mathbb{P}\left(\tau=\tau^{1}\right) .

Example 8.4.4.2:

A useful example is the case where \(\mu \equiv-1\). In this case, we obtain that \(\mathbb{1}_{\left\{t