CIR equations with multivariate L\'evy noise

TL;DR

This paper studies affine term structure models driven by multivariate Le9vy noise, generalizing classical CIR models by characterizing the short rate equations with complex noise structures and establishing solution equivalences.

Contribution

It introduces a framework for CIR equations with multivariate Le9vy noise, characterizes their generators, and shows solution equivalences under mild conditions.

Findings

Characterized the generator of the short rate process with multivariate Le9vy noise.

Proved that solutions are equivalent to those driven by independent stable coordinates.

Generalized classical CIR results to multivariate Le9vy processes.

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.