A geometric answer to an open question of singular control with stopping

We solve a problem of singular stochastic control with discretionary stopping, suggested as an interesting open problem by Karatzas, Ocone, Wang and Zervos (2000), by providing suitable candidates for the moving boundaries in an unsolved parameter range. We proceed by identifying an optimal stopping problem with similar variational inequalities and inspecting its parameter-dependent geometry (in a sense going back to Dynkin (1965)), which reveals a discontinuity not previously exploited. We thus highlight the potential importance of this geometric information in both singular control and parameter-dependent optimal stopping.