Affine processes under parameter uncertainty

Abstract

We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call non-linear affine processes. This is done as follows: given a set of parameters for the process, we construct a corresponding non-linear expectation on the path space of continuous processes. By a general dynamic programming principle we link this non-linear 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 the parameter set. This non-linear affine process yields a tractable model for Knightian uncertainty, especially for modelling interest rates under ambiguity. We then develop an appropriate Ito-formula, the respective term-structure equations and study the non-linear versions of the Vasicek and the Cox-Ingersoll-Ross (CIR) model. Thereafter we introduce the non-linear Vasicek-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 the approach solves this modelling issue arising with negative interest rates.