The McKean stochastic game (MSG) is a two-player version of a well-known optimal stopping problem which was first studied by McKean. The MSG consists of two players, together with a certain payoff function of an underlying random process. One player is looking for a strategy (stopping time) which minimises the expected pay-off, while the other player player tries to maximise this quantity.
For Brownian motion one can find the value of the MSG and the optimal stopping times by solving a free boundary value problem. It turns out that the optimal stopping region for the minimiser consists of a single point. For a Lévy process with jumps the corresponding free boundary problem seems more difficult to solve directly, and instead we use fluctuation theory to find the solution of the MSG driven by a spectrally negative Lévy process. We show in particular that, under some conditions on the characteristics of the Lévy process, the stopping region for the minimiser is an interval instead of a point. This talk is based on joint work with Andreas Kyprianou (University of Bath).