In the past years, deep learning algorithms have been applied to numerous classical problems from mathematical finance. In particular, deep learning has been employed to numerically solve high-dimensional derivatives pricing and hedging tasks. Theoretical foundations of deep learning for these tasks, however, are far less developed. In this talk, we start by revisiting deep hedging and introduce a recently developed adversarial training approach for making it more robust. We then present our recent results on theoretical foundations for approximating option prices, solutions to jump-diffusion PDEs and optimal stopping problems using (random) neural networks, allowing to obtain more explicit convergence guarantees. We address neural network expressivity, highlight challenges in analysing optimization errors and show the potential of random neural networks for mitigating these difficulties.

The Deligne-Beilinson conjecture predicts that the special values of many L-functions are related to the ranks of certain Ext groups in the category of mixed Hodge structures. In this talk, we present Skinner’s constructions of certain extensions that are extracted from the cohomology of the modular curve using CM points and the Eisenstein series. Through an explicit analytic calculation, which is performed in the adelic setting using (g,K)-cohomology and Tate’s zeta integrals, we obtain a formula relating the non-triviality of these extensions to the well-known non-vanishing at s=1 of the L-functions associated to Hecke characters of imaginary quadratic fields. These constructions have natural analogs in the category of p-adic Galois representations which are useful for Euler systems.