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

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.

You can condition on the value of $$\lambda_t$$. So
$$E[dN_t] = E[E[dN_t|\lambda_t]] = E[\lambda_t dt] = E[\lambda_t] dt$$