Martingale representation for Poisson processes with applications to minimal variance hedging

We consider a Poisson process ηη on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure λλ of ηη. We give a Clark–Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with ηη), which was previously known only in the special case, when λλ is the product of Lebesgue measure on R+R+ and a σσ-finite measure on another space XX. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of Itô of pure jump type and show that the Clark–Ocone type representation leads to an explicit version of the Kunita–Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.