Polynomial diffusions on compact quadric sets

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:127

Article

Larsson, Martin¹  Pulido, Sergio² 

[1] ETH, Dept Math, Ramistr 101, CH-8092 Zurich, Switzerland

[2] Univ Evry Val Essonne, ENSIIE, UMR CNRS 8071, Lab Math & Modelisat Evry LaMME, IBGBI 23 Blvd France, F-91037 Evry, France

关键词: Polynomial diffusion;    Sums of squares;    Biquadratic forms;    Stochastic invariance;    Pathwise uniqueness;    Smooth densities;   

DOI  :  10.1016/j.spa.2016.07.004

来源: Elsevier

【 摘 要 】

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 <= 4 every such biquadratic form is a sum of squares, while for d >= 6 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. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free