Measure-valued affine and polynomial diffusions

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 and obtain a representa- tion analogous to polynomial diffusions on Rm+ , 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 as it allows to transfer the most famous finite dimensional models, such as the Black-Scholes model, to an infinite dimensional measure-valued setting. We outline in particular the applicability of our approach for term structure modeling in energy markets.