Quasi-variational inequalities with Dirichlet boundary condition related to exit time problems for impulse control

We study degenerate-elliptic quasi-variational inequalities with Dirichlet boundary condition, which are related to the value function of the exit time problem for stochastic impulse control by means of the dynamic programming principle. The boundary condition in the viscosity solutions sense does not identify a unique solution, because in this nonlocal problem the boundary layer gives rise to a loss of information also at the interior points. The eventual discontinuities of solutions at the boundary of the domain play an essential role and cannot be removed. Therefore we superimpose a selection criterion which, enforcing the information coming from the boundary datum, picks up the value function among all possible viscosity solutions. As a result, we attain the continuity of the value function up to the boundary. In addition, we produce a monotone iterative scheme approximating the value function.