The Winter School on 'Theory and Practice of Optimal Stopping and Free Boundary Problems' was held in the School of Mathematics at the University of Leeds, 13-17 January 2020. The School was organised by Tiziano De Angelis, who is lecturer in financial/actuarial mathematics at Leeds, and it was funded by EPSRC via the First Grant EP/R021201/1. The school was attended by about 45 PhD students, post-docs and junior academics arriving from universities in the UK and overseas.
Invited lecturers were Erik Ekstrom (University of Uppsala), Damien Lamberton (Université Paris-Est – Marne-la-Vallée) and Mihalis Zervos (London School of Economics). The aim of the winter school was to provide a broad overview of mathematical methods for problems of optimal stopping and to illustrate some of the main applications of the theory. Optimal stopping is a very popular topic in stochastic calculus since the 1960s and it is used to model situations in which one or more individuals, who are subject to randomness, must choose the time to make a one-off decision which will produce a certain reward (eg, timing an investment opportunity, etc). Although the problem formulation is entirely probabilistic, there is a deep connection between optimal stopping theory and mathematical analysis, via the theory of so-called free boundary problems.
The series of lectures by Lamberton addressed the traditional analytical approach based on partial differential equations and variational inequalities. In his lectures, Lamberton made use of functional analysis and semi-group theory. Ekstrom instead presented an overview of solution methods for games of optimal stopping, based mostly on the geometric construction of the equilibrium strategies of the players. In his final lecture, Ekstrom also discussed the role of information in stopping games (eg, what happens if one player is more informed than the others). Finally, the series of lectures by Mihalis Zervos, gave an accessible introduction to free boundary problems arising from the mathematical formulation of so-called principal-agent problems. The latter have attracted great attention in economics and finance in the context of the optimal design of contracts between two parties (a 'principal' and an 'agent').
Throughout the week, students of the winter school gave 20-minute presentations on their research activity (there were 20 contributed talks). In the afternoon of the final day, De Angelis ran a three-hour tutorial class covering the solutions to selected question posed by the lecturers in their lectures.