We study an optimal dividend problem under a bankruptcy constraint where firms face a trade-off between potential bankruptcy and extraction of owner profits. In contrast to previous works, more 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 Hamilton--Jacobi--Bellman equation, and study 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, such as equity issuance and lotteries.
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