Variational inequalities on unbounded domains for zero-sum singular-controller vs. stopper games

We study a class of zero-sum games between a singular-controller and a stopper over finite-time horizon. The underlying process is a multi-dimensional (locally non-degenerate) controlled stochastic differential equation (SDE) evolving in an unbounded domain. We prove that such games admit a value and provide an optimal strategy for the stopper. The value of the game is shown to be the maximal solution, in a suitable Sobolev class, of a variational inequality of `min-max' type with obstacle constraint and gradient constraint. Although the variational inequality and the game are solved on an unbounded domain we do not require boundedness of either the coefficients of the controlled SDE or of the cost functions in the game.