Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 26J / iii / Solution

Put $L_t={\int }_0^t\Lambda (u)du$, finite almost surely. Conditional on the intensity process, $N(t)$ is Poisson with mean and variance $L_t$. The law of total expectation and law of total variance giveConsequently for a Cox process,These identities apply when the moments are finite; if the mean is finite but the integrated intensity has infinite variance, the count variance is infinite and the inequality still holds.