On the reducibility of affine models with dependent L\'evy factor

TL;DR

This paper investigates the structure of affine short rate models driven by dependent multivariate Lévy processes, showing they can be reduced to models involving one-dimensional stable Lévy processes under certain conditions.

Contribution

It generalizes classical CIR models to dependent multivariate Lévy processes and provides a classification framework for such affine models.

Findings

Identifies the generator form for models with dependent Lévy processes.

Shows equivalence to models driven by one-dimensional stable Lévy processes.

Extends classical CIR results to more general dependent Lévy settings.

Abstract

The paper is devoted to the study of the short rate equation of the form $$ dR(t)=F(R(t)) dt +\sum_{i=1}^{d}G(R(t-))dZ_i(t)$$ with deterministic functions $F,G_1,...,G_d$ and a multivariate L\'evy 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 L\'evy 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 $$ d R(t)=(a R(t)+b) dt+C\cdot (R(t-))^{1/\alpha} dZ^{\alpha}(t), \quad R(0)=x, $$ with some constants $a,b,C$ and a one dimensional $\alpha$-stable L\'evy 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 L\'evy process. It is the starting point for the classification in the spirit of \cite{DaiSingleton} and \cite{BarskiLochowski} for the affine models with dependent L\'evy processes.