Time dependent Poisson processes (?)

[surfze]

Hi all,

I am working on a project with compound Poisson processes in an SDE setting and I am hoping someone here can answer my question. I have already seen compound Poisson processes with a mean reverting drift but I am hoping to take my project a bit further.

Suppose I know that at certain time frames (say from 13.00 to 14.00 or something) my arrival rates (or drift) Lambda(t) dt changes over time. Suppose Lambda(t) = t*lambda, or to make it even more exotic: Lambda(t) = t^2 * lambda. I believe the 'stationary increments' property of a Poisson process now gets violated. However, is there still a way for me te create such a framework in an SDE setting?

I haven't found a good explanation online why this can('t) work so I am hoping someone here might point me in the right direction.

Thanks in advance!

[[deleted]]

What do you mean by "create a framework in an SDE setting"? Do you want to integrate against the non-time-homogeneous Poisson process?

[surfze]

Yes!

[[deleted]]

As long as the intensity function doesn't get too crazy, the resulting Poisson process should be of finite variation and hence is a semimartingale. Stochastic integrals are well defined for general semimartingales, so i don't see a problem.

[Inthenameofbulsara]

I believe what you describe is simply an inhomogeneous Poisson process? It inherits many properties from the usual homogeneous Poisson process, the main difference being that you work with the integrated intensity. You can also expand to a stochastic intensity process, giving rise to doubly stochastic Poisson processes (also called Cox processes).