How to calculate the expectation of Poisson process when its intensity is also stochastic

Question

How to calculate the expectation of Poisson process $N_t$ when its intensity is also stochastic? Since when intensity $\lambda_t$ is non-random, then we have $$E[dN_t] = \lambda_tdt.$$ But how about the stochastic $\lambda_t?$ I have no idea to calculate it. You can simply assume $\lambda_t$ is a Gaussian process.

Answer 1

We have $\mathbb E[N_t\mid \lambda_t]=t\lambda_t$ and to find expectation can apply: $$\mathbb EN_t=\mathbb E[\mathbb E[N_t\mid \lambda_t]]=\mathbb Et\lambda_t=t\mathbb E\lambda_t$$