This thesis studies mean-field games (MFGs) models of price formation. The thesis focuses explicitly on a MFGs price formation model proposed by Gomes and Saúde. The thesis is divided into two parts. The first part examines the deterministic supply case, while the second part extends the model to incorporate a stochastic supply function. We explore different approaches, such as Aubry-Mather theory, to study the properties of the MFGs price formation model and alternative formulations using a convex variational problem with constraints. We propose machine-learning-based numerical methods to approximate the solution of the MFGs price formation model in the deterministic and stochastic setting.
Julian received both B.S. in Mathematics and B.S. in Engineering from Escuela Colombiana de Ingenieria, Colombia. After receiving his M.Sc. in Mathematical Sciences from Universidad Nacional Autónoma de México, México, he joined Professor Diogo Gomes as a Ph.D. student in applied mathematics at King Abdullah University of Science and Technology, Saudi Arabia. His research interests are primarily optimal control, optimal transport, calculus of variations, partial differential equations, and mean-field games.