论文标题
带有多元Lévy噪声的CIR方程
CIR equations with multivariate Lévy noise
论文作者
论文摘要
本文专门研究$$ d r（t）= f（r（t））dt+sum_ {i = 1}^{d} g_i（r（t-））dz_i（t），\ quad r（quad r（quad r（0）= x \ ge_ quad t> quad t> g _多变量lévy过程$ z =（z_1，...，z_d）$。该方程式应该具有生成仿射项结构模型的非负解决方案。考虑了两类的噪声。在第一个中，Z的坐标是独立的过程，具有定期变化的拉普拉斯指数。在第二类Z中是一个球形过程，这意味着其Lévy度量的结构与稳定过程的结构相似，但具有一般形式的径向部分。对于这两个类别，均表征了短率发生器的精确形式。在温和的假设下，表明所考虑类型的任何方程都具有与具有独立稳定坐标的Lévy过程驱动的方程相同的解决方案。 本文将经典结果概括为Cox-Ingersoll-Ross（CIR）模型以及其扩展版本，其中$ z $是一维lévy流程。
The paper is devoted to the study of the short rate equation of the form $$ 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 Lévy 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 Lévy 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 Lévy 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 Lévy process.
