In this paper, we consider a company whose assets and liabilities evolve according to a correlated bivariate geometric Brownian motion, such as in Gerber and Shiu [(2003). Geometric Brownian motion models for assets and liabilities: From pension funding to optimal dividends. North American Actuarial Journal 7(3), 37–56]. We determine what dividend strategy maximises the expected present value of dividends until ruin in two cases: (i) when shareholders won't cover surplus shortfalls and a solvency constraint [as in Paulsen (2003). Optimal dividend payouts for diffusions with solvency constraints. Finance and Stochastics 7(4), 457–473] is consequently imposed and (ii) when shareholders are always to fund any capital deficiency with capital (asset) injections. In the latter case, ruin will never occur and the objective is to maximise the difference between dividends and capital injections. Developing and using appropriate verification lemmas, we show that the optimal dividend strategy is, in both cases, of barrier type. Both value functions are derived in closed form. Furthermore, the barrier is defined on the ratio of assets to liabilities, which mimics some of the dividend strategies that can be observed in practice by insurance companies. The existence and uniqueness of the optimal strategies are shown. Results are illustrated.

在本文中，我们考虑一家公司，其资产和负债根据相关的双变量几何布朗运动演化，例如 Gerber 和 Shiu [(2003)]。资产和负债的几何布朗运动模型：从养老金到最优股息。北美精算杂志 7(3), 37–56]。我们确定在两种情况下什么股息策略最大化股息的预期现值直到破产：（i）当股东不会弥补盈余短缺和偿付能力约束时[如 Paulsen（2003 年）。具有偿付能力约束的扩散的最佳股息支付。Finance and Stochastics 7(4), 457–473] 因此被强制执行并且 (ii) 当股东总是通过资本（资产）注入来弥补任何资本不足时。在后一种情况下，破产永远不会发生，目标是最大化股息和注资之间的差异。开发和使用适当的验证引理，我们表明，在这两种情况下，最优股息策略都是障碍类型的。两个价值函数都是以封闭形式导出的。此外，障碍是根据资产与负债的比率来定义的，它模仿了保险公司在实践中可以观察到的一些股息策略。显示了最优策略的存在性和唯一性。结果如图所示。障碍是根据资产与负债的比率来定义的，它模仿了保险公司在实践中可以观察到的一些股息策略。显示了最优策略的存在性和唯一性。结果如图所示。障碍是根据资产与负债的比率来定义的，它模仿了保险公司在实践中可以观察到的一些股息策略。显示了最优策略的存在性和唯一性。结果如图所示。