  This article investigates the use of random feature neural networks for
learning Kolmogorov partial (integro-)differential equations associated to
Black-Scholes and more general exponential L\'evy models. Random feature neural
networks are single-hidden-layer feedforward neural networks in which only the
output weights are trainable. This makes training particularly simple, but (a
priori) reduces expressivity. Interestingly, this is not the case for
Black-Scholes type PDEs, as we show here. We derive bounds for the prediction
error of random neural networks for learning sufficiently non-degenerate
Black-Scholes type models. A full error analysis is provided and it is shown
that the derived bounds do not suffer from the curse of dimensionality. We also
investigate an application of these results to basket options and validate the
bounds numerically.
  These results prove that neural networks are able to \textit{learn} solutions
to Black-Scholes type PDEs without the curse of dimensionality. In addition,
this provides an example of a relevant learning problem in which random feature
neural networks are provably efficient.
