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 li...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2021-12
Main Authors: Cuchiero, Christa, Guida, Francesco, Luca di Persio, Svaluto-Ferro, Sara
Format: Article
Language:English
Subjects:
Polynomials
Representations
Stochastic models
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 representation analogous to polynomial diffusions on \(\mathbb{R}^m_+\), 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 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.
ISSN:2331-8422