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 Θ 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 Θ. This non-linear 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 non-linear versions of the Vasiček and the Cox-Ingersoll-Ross (CIR) model. Thereafter we introduce the non-linear 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 the approach solves this modelling issue arising with negative interest rates.

Machine Learning for Multiple Yield Curve Markets: Fast Calibration in the Gaussian Affine Framework

“…The Gaussian process regression approach naturally comes with a posteriori distribution which contains much more information compared to the simple prediction (which contains only the mean). It seems to be highly interesting to utilize this for assessing the model risk of the calibration and compare it to the non-linear approaches recently developed in Fadina et al (2019) and in Hölzermann (2020).…”

Default Ambiguity

“…This paper provides a first step towards including ambiguity in intensity based models for credit risk. Many research questions are still open: first, the extension of constant boundaries λ, λ to time-dependent, or, as in Fadina et al (2019), state-dependent boundaries. Second, the extension to two or more defaultable assets, where default dependence comes into play.…”

“…It is of course possible to consider an ambiguity setting more general than the specific one in (6). One possibility is to consider only a subset of P. Another possibility is to allow the bounds λ and λ to depend on time, or even on the state of the process -this latter case is important for considering affine processes under uncertainty and we refer to Fadina et al (2019) for further details. In Section 5, we consider indeed such a more general setting.…”

“…The setting proposed above can, in dimension one, be directly linked to a special case of the non-linear affine processes introduced in Fadina et al (2019). Indeed, note that for a progressive process λ, the integral…”

“…The major advantage of this setting is that numerical methods via non-linear PDE come into reach. More precisely, Theorem 4.1 in Fadina et al (2019) shows that whenever ψ is Lipschitz, the non-linear expectation…”