Affine processes under parameter uncertainty

Probability, Uncertainty and Quantitative Risk

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[1] 0000 0001 2224 0361, grid.59025.3b, Nanyang Technological University, Division of Mathematical Sciences, Singapore, Singapore;grid.5963.9, Department of Mathematical Stochastics, University of Freiburg, Ernst-Zermelo Str.1, 79104, Freiburg, Germany;grid.5963.9, Department of Mathematical Stochastics, University of Freiburg, Ernst-Zermelo Str.1, 79104, Freiburg, Germany;grid.5963.9, Freiburg Institute of Advanced Studies (FRIAS), Freiburg im Breisgau, Germany;0000 0001 2157 9291, grid.11843.3f, University of Strasbourg Institute for Advanced Study (USIAS), Strasbourg, France;

关键词: Affine processes;    Knightian uncertainty;    Riccati equation;    Vasiček model;    Cox–Ingersoll–Ross model;    Nonlinear Vasiček/CIR model;    Heston model;    Itô formula;    Kolmogorov equation;    Fully nonlinear PDE;    Semimartingale;   

DOI  :  10.1186/s41546-019-0039-1

来源: publisher

【 摘 要 】

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.

【 授权许可】

CC BY