Polynomial diffusions on compact quadric sets

TL;DR

This paper characterizes polynomial diffusions on compact quadric sets, providing conditions for their existence and uniqueness, and explores the algebraic structure of biquadratic forms related to sum of squares representations.

Contribution

It offers a comprehensive analysis of polynomial diffusions on compact quadric sets, linking their properties to classical polynomial nonnegativity problems and establishing new results in dimensions up to four.

Findings

Every biquadratic form in dimension ≤ 4 is a sum of squares.

Counterexamples exist for biquadratic forms in dimension ≥ 6.

Conditions for existence, uniqueness, and boundary behavior of polynomial diffusions are established.

Abstract

Polynomial processes are defined by the property that conditional expectations of polynomial functions of the process are again polynomials of the same or lower degree. Many fundamental stochastic processes, including affine processes, are polynomial, and their tractable structure makes them important in applications. In this paper we study polynomial diffusions whose state space is a compact quadric set. Necessary and sufficient conditions for existence, uniqueness, and boundary attainment are given. The existence of a convenient parameterization of the generator is shown to be closely related to the classical problem of expressing nonnegative polynomials---specifically, biquadratic forms vanishing on the diagonal---as a sum of squares. We prove that in dimension $d\le 4$ every such biquadratic form is a sum of squares, while for $d\ge6$ there are counterexamples. The case $d=5$ remains open. An equivalent probabilistic description of the sum of squares property is provided, and we show how it can be used to obtain results on pathwise uniqueness and existence of smooth densities.