# Nonhomogenous Poisson Process

I have a question regarding the nonhomogenous Poisson process. From the definition I understand that $\lambda(s)$, the intensity of the process, is a deterministic function and the increments of the process $N_T-N_t \sim Pois\int_t^T \lambda(s)ds$. Does this hold true even for a generalisation where the intensity is assumed to be nondeterministic e.g. some jump diffusion process?

• If $\lambda$ is a random intensity, then the marginal distribution of $N_{t+s}-N_t$ no longer needs to be Poisson. – Sangchul Lee Jun 7 '18 at 8:14
• Are there conditions under which it still would be Poisson or can you point me please to a good resource? I am following this paper that discusses Hawkes process that seems to be using this property. Thanks – Michael Mark Jun 7 '18 at 8:28
• I know not much neither, but the Poisson process with random intensity is called Cox process and perhaps this keyword may be helpful. – Sangchul Lee Jun 7 '18 at 23:47
• Another term used in the literature is "doubly stochastic Poisson process." – Math1000 Jun 12 '18 at 7:29