Functional calculus and martingale representation formula for integer-valued measures

We construct a pathwise calculus for functionals of integer-valued measures and use it to derive an martingale representation formula with respect to a large class of integer-valued random measures. Using these results, we extend the Functional It\^o Calculus to functionals of integer-valued random measures. We construct a 'stochastic derivative' operator with respect to an integer-valued random measure, and show it to be the inverse of the stochastic integral with respect to the compensated measure. This stochastic derivative yields an explicit martingale representation formula for square-integrable martingales. Our results extend beyond the class of Poisson random measures and allow for random and time-dependent compensators.