Variational Analysis of Functionals of Poisson Processes

Let F(Π) be a functional of a (generally nonhomogeneous) Poisson process Π with intensity measure μ. Considering the expectation E_μF(Π) as a functional of μ from the cone 𝕄 of positive finite measures, we derive closed form expressions for its Fréchet derivatives of all orders that generalize the perturbation analysis formulae for Poisson processes. Variational methods developed for the space 𝕄 allow us to obtain first and second order sufficient conditions for various types of constrained optimization problems for E_μF. 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.