Finite-time horizon, stopper vs. singular-controller games on the half-line

We prove existence of a value for two-player zero-sum stopper vs. singular-controller games on finite-time horizon, when the underlying dynamics is one-dimensional, diffusive and bound to evolve in $[0,\infty)$. 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 $[0,T)\times\{0\}$. Moreover, we obtain an optimal strategy for the stopper. Compared to the existing literature on this topic, we introduce new probabilistic methods to obtain gradient bounds and equi-continuity for the solutions of penalised partial differential equations (PDE) that approximate the variational inequality.