Andrea Bovo (University of Turin) – An overview on recent results on Stopper vs. Singular-controller games

We study various formulation of zero-sum games between a singular-controller and a stopper with a finite-time horizon, where the underlying process is a multi-dimensional controlled stochastic differential equation evolving in an unbounded domain. We prove that such games admit a value and present an optimal strategy for the stopper. In some cases, we show the game's value is the maximal solution, in a suitable Sobolev class, of a variational inequality of 'min-max' type with both obstacle and gradient constraint. Under stricter assumptions, we provide an optimal strategy for the controller and establish a connection between the space derivative of the value function and the solution of an optimal stopping problem with absorption.