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

The paper is devoted to the study of the short rate equation of the form dR(t)=F(R(t)) dt +∑ᵢ₌₁ᵈG(R(t-))dZᵢ(t) with deterministic functions F,G₁,...,Gd and a multivariate L\'evy process Z=(Z₁,...,Zd) with possibly dependent coordinates. The equation is supposed to have a nonnegative solution which generates an affine term structure model. The L\'evy measure ν of Z is assumed to admit a spherical decomposition based on the representation Rᵈ=Sᵈ⁻¹× (0,+∞), where Sᵈ⁻¹ stands for the unit sphere. Then ν(dy)=λ(dξ)× γξ(dr), where λ is a measure on Sᵈ⁻¹ and γξ on (0,+∞). 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· (R(t-))1/α dZα(t), R(0)=x, with some constants $a,b,C$ and a one dimensional α-stable L\'evy process Zα, where α∈(1,2). The case when ν has a density is considered as a special case. The paper generalizes the classical results on the Cox-Ingersoll-Ross (CIR) model, {CIR}, as well as on its extended version from {BarskiZabczykCIR} and {BarskiZabczyk} where Z is a one-dimensional L\'evy process. It is the starting point for the classification in the spirit of {DaiSingleton} and {BarskiLochowski} for the affine models with dependent L\'evy processes.