We study an optimal dividend problem under a bankruptcy constraint. Firms face a trade-off between potential bankruptcy and extraction of profits. In contrast to previous works, general cash flow drifts, including Ornstein–Uhlenbeck and CIR processes, are considered. We provide rigorous proofs of continuity of the value function, whence dynamic programming, as well as uniqueness of the solution to the Hamilton–Jacobi–Bellman equation, and study its qualitative properties both analytically and numerically. The value function is thus given by a nonlinear PDE with a gradient constraint from below in one dimension. We find that the optimal strategy is both a barrier and a band strategy and that it includes voluntary liquidation in parts of the state space. Finally, we present and numerically study extensions of the model, including equity issuance and gambling for resurrection.

The classical duality theory of Kantorovich [24] and Kellerer [25] for the classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided. This abstract generalization is set in a Banach lattice X with a unit order. The primal problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of X and the dual problem is defined on the bidual of X. These results are then applied to several extensions of the classical optimal transport. In particular, an alternate proof of Kellerer’s result is given without using the Choquet Theorem.