Doubly stochastic Poisson processes whose unobservable intensity is a shot-noise process with random amplitude arise when each event of a primary Poisson process generates a random number of subsidiary events. We derive a stochastic partial differential equation for the unnormalized conditional moment generating function. This equation can be used for recursive compution of the minimum variance estimator of the unobservable intensity as well as the likelihood ratio with respect to the reference measure, on the basis of point process observations.