独立Lévy过程驱动的仿射模型的分类与标定
Classification and calibration of affine models driven by independent Lévy processes
摘要：
本文研究了具有确定性函数$F,G_1,...,G_d$和独立的无穷变分Lévy过程$Z_1,...,Z_d$且Laplace指数规则变化的$$ dR(t)=F(R(t))dt+\sum_{i=1}^{d}G_i(R(t-))dZ_i(t), \quad R(0)=x\geq 0, \quad t>0, $$形式的短速率方程.假设该方程有一个非负解，从而得到一个仿射期限结构模型。一个精确的形式的生成器的$R$的特点和相关的分类方程生成仿射模型介绍戴和辛格尔顿\cite{DaiSingleton}的精神。每个类都有自己的规范表示，这是一个方程具有相同的漂移和跳跃扩散部分的基础上的Lévy过程采取值在$\mathbb{R}^{g}, 1\leq g\leq d$，独立的坐标是稳定的过程与稳定指数的范围$(1,2]$。给出了市场利率期限结构正则表达式的数值校准结果，并与经典的CIR模型进行了比较。本文推广了\cite{CIR}关于CIR模型的经典结果，以及\cite{BarskiZabczykCIR}和\cite{BarskiZabczyk}关于CIR模型的推广结果，其中$Z$是一维Lévy过程.
英文摘要：
The paper is devoted to the study of the short rate equation of the form $$ dR(t)=F(R(t))dt+\sum_{i=1}^{d}G_i(R(t-))dZ_i(t), \quad R(0)=x\geq 0, \quad t>0, $$ with deterministic functions $F,G_1,...,G_d$ and independent Lévy processes of infinite variation $Z_1,...,Z_d$ with regularly varying Laplace exponents. The equation is supposed to have a nonnegative solution which generates an affine term structure model. A precise form of the generator of $R$ is characterized and a related classification of equations which generate affine models introduced in the spirit of Dai and Singleton \cite{DaiSingleton}. Each class is shown to have its own canonical representation which is an equation with the same drift and the jump diffusion part based on a Lévy process taking values in $\mathbb{R}^{g}, 1\leq g\leq d$, with independent coordinates being stable processes with stability indices in the range $(1,2]$. Numerical calibration results of canonical representations to the market term structure of interest rates are presented and compared with the classical CIR model. The paper generalizes the classical results on the CIR model from \cite{CIR}, as well as on its extended version from \cite{BarskiZabczykCIR} and \cite{BarskiZabczyk} where $Z$ was a one-dimensional Lévy process.
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