The McKean stochastic game driven by a spectrally negative Levy process

The McKean stochastic game (MSG) is a two-player version of the perpetual American put option. The MSG consists of two agents and a certain payoff function of an underlying stochastic process. One agent (the seller) is looking for a strategy (stopping time) which minimises the expected pay-off, while the other agent (the buyer) 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. For a Lévy process with jumps the corresponding free boundary problem is more difficult to solve directly and instead we use fluctuation theory to find the solution of the MSG driven by a Lévy process with no positive jumps. One interesting aspect is that the optimal stopping region for the minimiser "thickens" from a point to an interval in the presence of jumps. This talk is based on joint work with Andreas Kyprianou (University of Bath).