Measure-valued affine and polynomial diffusions

Abstract

We introduce a class of measure-valued processes, which – in analogy to their finite dimensional counterparts – will be called measure-valued polynomial diffusions. We show the so-called moment formula, i.e. a representation of the conditional marginal moments via a system of finite dimensional linear PDEs. Furthermore, we characterize the corresponding infinitesimal generators obtaining a representation analogous to polynomial diffusions on $R_{+}^m$, in cases where their domain is large enough. In general the infinite dimensional setting allows for richer specifications strictly beyond this representation. As a special case, we recover measure-valued affine diffusions, sometimes also called Dawson–Watanabe superprocesses. From a mathematical finance point of view, the polynomial framework is especially attractive since it allows to transfer many famous finite dimensional models and their tractability properties to an infinite dimensional measure-valued setting.