Functional Ito-Calculus for Superprocesses and the Historical Martingale Representation

Abstract

We derive the functional Ito-formula for Dawson-Watanabe superprocesses, a well-known class of measure-valued processes. In addition, we show that by extending the functional derivative used in the functional Ito-formula we obtain the integrand in the martingale representation formula for square-integrable F_t-martingales. In this case, F_t is the filtration generated by an underlying superprocess. This result is finally extended to square-integrable historical martingales, i.e. square-integrable H_t-martingales with H_t being the filtration generated by a historical Brownian motion.