  In this work we consider one-dimensional generalized affine processes under
the paradigm of Knightian uncertainty (so-called non-linear generalized affine
models). This extends and generalizes previous results in Fadina et al. (2019)
and L\"utkebohmert et al. (2022). In particular, we study the case when the
payoff is allowed to depend on the path, like it is the case for barrier
options or Asian options. To this end, we develop the path-dependent setting
for the value function which we do by relying on functional It\^o calculus. We
establish a dynamic programming principle which then leads to a functional
non-linear Kolmogorov equation describing the evolution of the value function.
While for Asian options, the valuation can be traced back to PDE methods, this
is no longer possible for more complicated payoffs like barrier options. To
handle such payoffs in an efficient manner, we approximate the functional
derivatives with deep neural networks and show that the numerical valuation
under parameter uncertainty is highly tractable. Finally, we consider the
application to structural modelling of credit and counterparty risk, where both
parameter uncertainty and path-dependence are crucial and the approach proposed
here opens the door to efficient numerical methods in this field.
