Variational analysis of functionals of Poisson processes

Let F(II) be a functional of a (generally nonhomogeneous) Poisson process II with intensity measure mu. Considering the expectation EmuF(II) as a functional of mu from the cone M of positive finite measures, we derive closed form expressions for its Frechet derivatives of an orders that generalize the perturbation analysis formulae for Poisson processes. Variational methods developed for the space mm allow us to obtain first and second order sufficient conditions for various types of constrained optimization problems for EmuF. We study in detail optimization over the class of measures with a fixed total mass a and develop a technique that often allows us to obtain the asymptotic behavior of the optimal intensity measure in the high intensity setting when a grows to infinity. As a particular application we consider the problem of optimal placement of stations in the Poisson model of a two-layer telecommunication network.