Finite-Time Horizon, Stopper vs. Singular-Controller Games on the Half-Line

We prove the existence of a value for two-player zero-sum stopper versus singular-controller games on a finite-time horizon when the underlying dynamics are one-dimensional, diffusive and bound to evolve in [0, infinity). We show that the value is the maximal solution of a variational inequality with both obstacle and gradient constraint and satisfying a Dirichlet boundary condition at [0, T) x {0}. Moreover, we obtain an optimal strategy for the stopper. In order to achieve our goals, we rely on new probabilistic methods, yielding gradient bounds and equicontinuity for the solutions of penalised partial differential equations that approximate the variational inequality.