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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.

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You can condition on the value of $\lambda_t$. So

$E[dN_t] = E[E[dN_t|\lambda_t]] = E[\lambda_t dt] = E[\lambda_t] dt$

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