Title: 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.