This 6-hour course covers the theory of optimal control in the case of discrete spaces / graphs. In the first part, we present the dynamic programming principle and the resulting Bellman equations. Bellman equations, which turn out to be a system of backward ordinary differential equations (ODE), are then thoroughly studied: in addition to existence and uniqueness results obtained through classical ODE tools and comparison principles, the long-term behavior of optimal control problems is studied using comparison principles and semi-group tools.
The second part of the course focuses on a special case of optimal control problems on graphs for which closed-form solutions can be derived. The link with inventory management problems will be presented in details (in particular the link with the resolution of the Avellaneda-Stoikov problem, a classical problem in finance).
Olivier Guéant is Full Professor of Applied Mathematics at Université Paris 1 Panthéon-Sorbonne and Adjunct Professor of Finance at ENSAE – IP Paris where he carried out research on optimal control, mean field games, and quantitative finance.
Olivier Guéant studied mathematics and economics at Ecole Normale Supérieure and was a special student at Harvard University. He defended the first PhD thesis on mean field games under the supervision of Pierre-Louis Lions. Guéant's PhD was awarded the Rosemont Demassieux prize. In 2010, he founded with Jean-Michel Lasry, Pierre-Louis Lions, and Henri Verdier a start-up called MFG-Labs, pioneering in Big Data. The company has been acquired in 2013 by a global marketing and communications group. Prior to joining Université Paris 1 Panthéon-Sorbonne in 2016, he was associate professor of applied mathematics at Université Paris Diderot from 2010 to 2015 and then Professor of Quantitative finance at ENSAE from 2015 to 2016.