Differential calculus relative to some point processes

Abstract

The differential calculus on Wiener space (respectively on Poisson space) is based essentially on some sort of gradient operator and divergence operator in infinite dimension. The relation of duality between these two operators provides the Wiener and Poisson spaces with an integration by parts formula. This formula plays a central role in the stochastic calculus of variations. Our aim in this paper is to develop an analogous calculus on a point process with compensator ∫ 0 t ds Φ(s) where Φ may depend predictably upon the whole past.