Deep learning algorithms have been shown empirically to work well in many classical problems from mathematical finance. Theoretical foundations of deep learning in this context, however, are far less developed. In this talk we present our recent results on error guarantees for approximating option prices, solutions to jump-diffusion PDEs and optimal stopping problems using (random) neural networks. We address neural network expressivity, highlight challenges in analysing optimization and show how random neural networks are able to mitigate these difficulties. Thereby, randomization yields a fully-implementable neural network-based learning algorithm that provably overcomes the curse of dimensionality in certain practically relevant situations.
