Can jump intensity be unknown at time t?

Question

I would like to define something akin to a Poisson process but where the jump intensity is a random variable not known at time t. Is this possible? Are there references about this? The ultimate goal is to describe a system of N particles evolving, for $1 \leq i \leq N$ as $$ dx_{it} = J_{it}dN_{it}\\ $$ where $N_{it}$ is Poisson process with intensity given by $\lambda_t$, drawn every period, so that, for instance, $$E[dx_{it}] = E[J_{it}]E[\lambda_t]dt$$

Answer 1

What is known or not known is usually encoded in filtrations. Roughly speaking, if the counting process $N_{it}$ is adapted to the filtration $\mathcal{F}_t$, it has an $\mathcal{F}_t$-compensator $\Lambda_{it}$. If that is absolutely continuous, we may talk about the $\mathcal{F}$-intensity $\lambda_{it}$ with the property $ \Lambda_{it} = \int_0^t \lambda_{is} ds $. You may then define another process $X_{it}$ based on $N_{it}$ as you describe.

However, an observer might only see $X_{it}$. In terms of $\sigma$-algebras, this means that he is only able to use information encoded in $\mathcal{F}^{X_i}_t$. Since $\lambda_{it}$ is adapted to $\mathcal{F}_t$, but not $\mathcal{F}^{X_i}_t$, it cannot be computed from data in $\mathcal{F}^{X_i}_t$ alone. However, you may ask what is $\mathbb{E}[\lambda_{it}|\mathcal{F}^{X_i}_{t-}]$. This is also called $\mathcal{F}^{X_i}_t$-intensity of $N_{it}$ and describes the probability of an event given observations of the process $X_i$ up to (but not including) time $t$.