  We consider a dynamic system that is driven by an intensity-modulated Poisson
process with intensity $\Lambda(t)=\lambda(t)+\varepsilon\nu(t)$. We derive an
exact relation between the input-output cross-correlation in the spontaneous
state ($\varepsilon=0$) and the linear response to the modulation
($\varepsilon>0$). This can be regarded as a variant of the Furutsu-Novikov
theorem for the case of shot noise. As we show, the relation is still valid in
the presence of additional independent noise. Furthermore, we derive an
extension to Cox-process input, i.e. to colored shot noise. We discuss
applications to particle detection and to neuroscience. Using the new relation,
we obtain a fluctuation-response-relation for a leaky integrate-and-fire
neuron. We also show how the new relation can be used in a remote control
problem in a recurrent neural network. The relations are numerically tested for
both stationary and non-stationary dynamics. Lastly, extensions to marked
Poisson processes and to higher-order statistics are presented.
