Machine Learning in Finance

Abstract: Mean field games have been introduced to study games with a very large number of players. They combine mean field approximation techniques borrowed from statistical physics to describe the population and optimal control techniques to describe the behavior of a representative player. Mean field games and control problems have found many applications, particularly in economics and finance. Master equations are partial differential equations introduced by Pierre-Louis Lions to characterize Nash equilibria in such games. The unknown is a function taking the population distribution as an input, which is in general high dimensional. This motivates the investigation of deep learning methods for such equations. These methods typically rely on three ingredients: (1) representation of the distribution, (2) approximation of the function of interest, and (3) training algorithm. Since we cannot train on every possible distribution, the question of generalisation is unavoidable. In this respect, deep neural networks seem to be a promising tool. We will describe approaches that are based on the full knowledge of the model, as well as approaches that are model-free.