Probability measure-valued polynomial diffusions

TL;DR

This paper introduces a new class of infinite-dimensional probability measure-valued diffusions called polynomial diffusions, extending finite-dimensional polynomial process properties and enabling tractable analysis via PDEs.

Contribution

It generalizes finite-dimensional polynomial processes to an infinite-dimensional setting, preserving tractability and providing new representations for generators and martingale problems.

Findings

Representation of conditional moments via finite-dimensional PDEs

Well-posedness and uniqueness of associated martingale problems

Extension of polynomial process properties to measure-valued diffusions

Abstract

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming--Viot process is a particular example. The defining property of finite dimensional polynomial processes considered by Cuchiero et al. (2012) and Filipovic and Larsson (2016) is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.