I am wondering if, considering $N_t$- an $\mathcal{F}_t$ poisson process with stochastic intensity $\lambda_t$ on $(\Omega,P)$ and $\tilde{N}_t$- an $\mathcal{G}_t$ poisson process with stochastic intensity $\lambda_t$ on $(\Omega,Q)$ then can we say that $N_t$ and $\tilde{N}_t$ have the same law?
2026-03-26 04:28:29.1774499309
Stochastic intensity poisson process
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The requirements to be a Poisson process do depend on the filtration. Roughly speaking, to be adapted is easier when the filtration is larger, and the condition involving predictable processes is more difficult to meet since there are more predictable processes when the filtration is larger. But all this is offtopic to determine whether the distribution of the process is Poisson or not: (1.) one needs to be sure that the process is adapted to the filtration, to take conditional expectations, and (2.) one needs to be able to compute the joint distributions of families of random variables taken from the process. These are both guaranteed by the fact that the process is Poisson (hence adapted, which gives (1.)) with given intensity function (which gives (2.)).