Further calculations for the McKean stochastic game for a spectrally negative Levy process: from a point to an interval

TL;DR

This paper refines the understanding of the McKean stochastic game driven by spectrally negative Levy processes, detailing how the exercise region changes with the penalty parameter and providing explicit formulas for different cases.

Contribution

It improves the characterization of the saddle point in the McKean stochastic game with Gaussian and negative jumps, showing the transition of the exercise region from a singleton to an interval.

Findings

Exercise region is a singleton when penalty exceeds a threshold.

Exercise region expands to an interval when penalty drops below the threshold.

Explicit formulas for scale functions and polynomials are provided.

Abstract

Following Baurdoux and Kyprianou [2] 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.