Random Feature Neural Networks Learn Black-Scholes Type PDEs Without Curse of Dimensionality
Lukas Gonon.
Year: 2023, Volume: 24, Issue: 189, Pages: 1−51
Abstract
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 Lévy 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 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.