A Functional Ito-Formula for Dawson-Watanabe Superprocesses

TL;DR

This paper develops a functional Ito-formula for Dawson-Watanabe superprocesses, extending classical results to infinite-dimensional state spaces and path-dependent functions, with implications for martingale representation.

Contribution

It introduces a novel functional Ito-formula for measure-valued superprocesses, extending the state space and incorporating path-dependent functions.

Findings

Derived an Ito-formula for Dawson-Watanabe superprocesses.

Extended the state space to infinite-dimensional measures.

Discussed applications to martingale representation.

Abstract

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