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Affine processes under parameter uncertainty
Probability, Uncertainty and Quantitative Risk Pub Date : 2019-05-28 , DOI: 10.1186/s41546-019-0039-1 Tolulope Fadina , Ariel Neufeld , Thorsten Schmidt
Probability, Uncertainty and Quantitative Risk Pub Date : 2019-05-28 , DOI: 10.1186/s41546-019-0039-1 Tolulope Fadina , Ariel Neufeld , Thorsten Schmidt
We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call nonlinear affine processes. This is done as follows: given a set Θ of parameters for the process, we construct a corresponding nonlinear expectation on the path space of continuous processes. By a general dynamic programming principle, we link this nonlinear expectation to a variational form of the Kolmogorov equation, where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in Θ. This nonlinear affine process yields a tractable model for Knightian uncertainty, especially for modelling interest rates under ambiguity. We then develop an appropriate Itô formula, the respective term-structure equations, and study the nonlinear versions of the Vasiček and the Cox–Ingersoll–Ross (CIR) model. Thereafter, we introduce the nonlinear Vasiček–CIR model. This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence this approach solves the modelling issue arising with negative interest rates.
中文翻译:
参数不确定性下的仿射过程
我们在参数不确定性下建立了仿射过程的一维概念,我们将其称为非线性仿射过程。这样做如下:给定过程参数的Θ,我们在连续过程的路径空间上构造相应的非线性期望。通过一般的动态规划原理,我们将此非线性期望与Kolmogorov方程的变分形式联系起来,其中仿射过程的所有相应生成器中的单个仿射过程的生成器均被替换为Θ中的参数。这种非线性仿射过程为Knightian不确定性提供了一个易于处理的模型,尤其是在模棱两可的情况下为利率建模时。然后,我们开发一个合适的Itô公式,相应的术语结构方程式,并研究Vasiček和Cox-Ingersoll-Ross(CIR)模型的非线性版本。此后,我们介绍了非线性Vasiček-CIR模型。当人们不想事先限制状态空间时,该模型特别适合于对利率建模,因此该方法解决了负利率引起的建模问题。
更新日期:2019-05-28
中文翻译:
参数不确定性下的仿射过程
我们在参数不确定性下建立了仿射过程的一维概念,我们将其称为非线性仿射过程。这样做如下:给定过程参数的Θ,我们在连续过程的路径空间上构造相应的非线性期望。通过一般的动态规划原理,我们将此非线性期望与Kolmogorov方程的变分形式联系起来,其中仿射过程的所有相应生成器中的单个仿射过程的生成器均被替换为Θ中的参数。这种非线性仿射过程为Knightian不确定性提供了一个易于处理的模型,尤其是在模棱两可的情况下为利率建模时。然后,我们开发一个合适的Itô公式,相应的术语结构方程式,并研究Vasiček和Cox-Ingersoll-Ross(CIR)模型的非线性版本。此后,我们介绍了非线性Vasiček-CIR模型。当人们不想事先限制状态空间时,该模型特别适合于对利率建模,因此该方法解决了负利率引起的建模问题。