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 the hidden weights are randomly generated and only the output
weights are trainable. This makes training particularly simple, but (a priori) reduces expressivity. Interestingly, this is not the case for certain 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 – bounding
the approximation, generalization and optimization error of the algorithm – 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 learn solutions to suitable BlackScholes 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.