The Shepp-Shiryaev stochastic game driven by a spectrally negative Lévy process

Abstract

In [A. E. Kyprianou, Finance Stoch., 8 (2004), pp. 73–86], the stochastic-game-analogue of Shepp and Shiryaev's optimal stopping problem (cf. [L. A. Shepp and A. N. Shiryaev, Ann. Appl. Probab., 3 (1993), pp. 631–640] and [L. A. Shepp and A. N. Shiryaev, Theory Probab. Appl., 39 (1994), pp. 103–119]) was considered when driven by an exponential Brownian motion. We consider the same stochastic game, which we call the Shepp–Shiryaev stochastic game, but driven by a spectrally negative Lévy process and for a wider parameter range. Unlike [A. E. Kyprianou, Finance Stoch., 8 (2004), pp. 73–86], we do not appeal predominantly to stochastic analytic methods. Principally, this is due to difficulties in writing down variational inequalities of candidate solutions on account of then having to work with nonlocal integro-differential operators. We appeal instead to a mixture of techniques including fluctuation theory, stochastic analytic methods associated with martingale characterizations, and reduction of the stochastic game to an optimal stopping problem