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 comparison between discontinuous sub‐ and supersolutions of the Hamilton–Jacobi–Bellman equation, and we provide an efficient and convergent numerical scheme for finding the solution. The value function is given by a nonlinear partial differential equation (PDE) with a gradient constraint from below in one direction. 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.

中文翻译：

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具有随机获利能力的最优股息政策

我们研究了破产约束下的最优股利问题。公司面临着潜在破产与利润提取之间的权衡。与以前的工作相比，考虑了包括Ornstein–Uhlenbeck和CIR流程在内的一般现金流量漂移。我们提供了值函数连续性，动态编程的严格证明，以及Hamilton-Jacobi-Bellman方程的不连续子解和超解之间的比较，并且我们提供了一种有效且收敛的数值方案来寻找该解。值函数由一个非线性偏微分方程（PDE）给出，该方程具有从一个方向上从下方向下的梯度约束。我们发现，最优策略既是障碍策略，也是带策略，并且它包括部分状态空间中的自愿清算。最后，