Functional calculus and martingale representation formula for integer-valued random measures
We develop a non-anticipative calculus for functionals of a continuous martingale and an integer-valued random measure. We first construct a non-anticipative functional derivative, which is shown to be the inverse of the stochastic integration for a dense functional class of integrands. We then show that this operator has a unique closure which is an isometry and defines a stochastic derivative acting on stochastic integrals. This stochastic derivative yields a constructive martingale representation formula and an orthogonal decomposition. The construction allows for random compensators, singular continuous time dependence and degenerate continuous covariation. On Wiener-Poisson space, the stochastic derivative is shown to be the predictable projection of the Malliavin derivative; we also obtain a factorization based on local mark gradients. We exhibit several applications of this framework to integer-valued random measures: sensitivity formulas and a convergent expansion for expectations under bounded predictable perturbations of the compensator, a change-of-measure formula for the derivative, a decomposition of the relative entropy in terms of the stochastic derivative, and a pathwise refinement of the Mecke formula.