On the reducibility of affine models with dependent Lévy factor

Abstract

The paper is devoted to the study of the short rate equation of the form $$dR(t)=F(R(t))dt+\sum _{i=1}^{d}G(R(t-))d{Z}_i(t)$$ with deterministic functions $F,G_1,...,G_d$ and a multivariate Lévy process $Z=(Z_1,...,Z_d)$ with possibly dependent coordinates. The equation is supposed to have a nonnegative solution which generates an affine term structure model. The Lévy measure $\nu$ of $Z$ is assumed to admit a spherical decomposition based on the representation ${\mathbb{R}}^d=S^{d-1}\times (0,+\infty )$, where $S^{d-1}$ stands for the unit sphere. Then $\nu (dy)=\lambda (d\xi )\times {\gamma }_{\xi }(dr)$, where $\lambda$ is a measure on $S^{d-1}$ and ${\gamma }_{\xi }$ on $(0,+\infty )$. Under some assumptions on spherical decomposition, a precise form of the generator of $R$ is determined and it is shown that the resulted term structure model is identical to that generated by the equation $$dR(t)=(aR(t)+b)dt+C\cdot (R(t-){)}^{1/\alpha }d{Z}^{\alpha }(t),R(0)=x,$$ with some constants $a,b,C$ and a one dimensional $\alpha$-stable Lévy process $Z^{\alpha }$, where $\alpha \in (1,2)$. The case when $\nu$ has a density is considered as a special case. The paper generalizes the classical results on the Cox-Ingersoll-Ross (CIR) model, \cite{CIR}, as well as on its extended version from \cite{BarskiZabczykCIR} and \cite{BarskiZabczyk} where $Z$ is a one-dimensional Lévy process. It is the starting point for the classification in the spirit of \cite{DaiSingleton} and \cite{BarskiLochowski} for the affine models with dependent Lévy processes.

arXiv admin note: text overlap with arXiv:2204.07245