The Wills functional for Poisson processes

This paper establishes a Poisson analogue of the Wills functional for Gaussian processes. For a Poisson process PtPt, t∈[0,T]t∈[0,T], with intensity function λtλt, let Xt=Pt−∫0tλsds. We derive an expression for W(P,r)=E{exp(max0≤t≤T[rXt−r22EXt2])},r≥0.