No abstract

A weak extension of the Dupire derivative is derived, which turns out to be the adjoint operator of the integral with respect to the martingale measure associated with the historical Brownian motion a benchmark example of a measure valued process. This extension yields the explicit form of the martingale representation of historical functionals, which we compare to a classical result on the representation of historical functionals derived in [7].

We derive an Itō-formula for the Dawson-Watanabe superprocess, a wellknown class of measure-valued processes, extending the classical Itō-formula with respect to two aspects. Firstly, we extend the state-space of the underlying process (X(t)) t∈[0,T ] to an infinite-dimensional one -the space of finite measure. Secondly, we extend the formula to functions F (t, Xt) depending on the entire paths Xt = (X(s ∧ t)) s∈[0,T ] up to times t. This later extension is usually called functional Itō-formula. Finally we remark on the application to predictable representation for martingales associated with superprocesses.