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, based on a compactness substitute for nondecreasing processes, the Meyer-Zheng weak convergence, and the maximum principle of Pontryagin, establishes existence under minimal conditions, produces general approximation results and further elucidates the celebrated connection between optimal stochastic control and stopping.