Existence, Characterization and Approximation in the Generalized Monotone-Follower Problem

We revisit the classical monotone-follower problem and consider it in a generalized formulation. Our approach is based on a compactness substitute for nondecreasing processes, the Meyer-Zheng weak convergence, and the maximum principle of Pontryagin. It establishes existence under weak conditions, produces general approximation results and further elucidates the celebrated connection between singular stochastic control and stopping.