Following Baurdoux and Kyprianou (2008) we consider the McKean stochastic game, a game version of the McKean optimal stopping problem (American put), driven by a spectrally negative Levy process. We improve their characterisation of a saddle point for this game when the driving process has a Gaussian component and negative jumps. In particular, we show that the exercise region of the minimiser consists of a singleton when the penalty parameter is larger than some threshold and 'thickens' to a full interval when the penalty parameter drops below this threshold. Expressions in terms of scale functions for the general case and in terms of polynomials for a specific jump diffusion case are provided.
- optimal stopping
- stochastic game
- fluctuation theory
- Levy process
Baurdoux, E. J., & Van Schaik, K. (2011). Further calculations for the McKean stochastic game for a spectrally negative Levy process: from a point to an interval. Journal of Applied Probability, 48(1), 200-216. https://doi.org/10.1239/jap/1300198145