We consider an optimal dividend and capital structure problem for a firm, which holds a certain amount of debt to which is associated a financial ratio covenant between the firm and its creditors. We study optimal policies under a bankruptcy framework, using a mixed reduced and structural approach in modeling default and liquidation times. Once in default, the firm is given a grace period during which it may inject more capital to correct the situation. The firm is liquidated if, by the end of the grace period, assets do not exceed the debt. Under this setup, we maximize the discounted amount of dividends distributed minus the capital injected up to the time of liquidation. It gives rise to a two-dimensional singular control problem leading to a non-standard system of variational inequalities. Beyond the usual viscosity characterization, we completely solve this problem and obtain a description of the continuation, dividend and capital injection regions enabling us to fully characterize the optimal policies. We conclude the paper with numerical results and illustrations.

中文翻译：

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具有债务契约的最佳股息和资本结构

我们考虑一家公司的最优股息和资本结构问题，该公司持有一定数量的债务，该债务与公司与其债权人之间的财务比率契约相关联。我们研究破产框架下的最优政策，使用混合简化和结构方法来模拟违约和清算时间。一旦违约，公司就会有一个宽限期，在此期间它可以注入更多资金来纠正这种情况。如果在宽限期结束时资产不超过债务，公司将被清算。在这种设置下，我们最大化分配的股息贴现量减去到清算时注入的资本。它产生了一个二维奇异控制问题，导致了一个非标准的变分不等式系统。除了通常的粘度表征之外，我们完全解决了这个问题，并获得了延续、分红和注资区域的描述，使我们能够充分表征最优政策。我们用数值结果和插图来结束本文。