CIR equations with multivariate L\'evy noise

Abstract

The paper is devoted to the study of the short rate equation of the form $$ d R(t)=F(R(t)) dt+\sum_{i=1}^{d}G_i(R(t-)) dZ_i(t), \quad R(0)=x\geq 0,\quad t>0, $$ with deterministic functions $F,G_1,...,G_d$ and a multivariate L\'evy process $Z=(Z_1,...,Z_d)$. The equation is supposed to have a nonnegative solution which generates an affine term structure model. Two classes of noise are considered. In the first one the coordinates of Z are independent processes with regularly varying Laplace exponents. In the second class Z is a spherical processes, which means that its L\'evy measure has a similar structure as that of a stable process, but with radial part of a general form. For both classes a precise form of the short rate generator is characterized. Under mild assumptions it is shown that any equation of the considered type has the same solution as the equation driven by a L\'evy process with independent stable coordinates. The paper generalizes the classical results on the Cox-Ingersoll-Ross (CIR) model as well as on its extended version where $Z$ is a one-dimensional L\'evy process.