扩散模型中的资本注入与绝对连续股息支付优化
Optimization of capital injections and absolutely continuous dividend payments in a diffusion model
查看机构详情
- Département de mathématiques, Université du Québec à Montréal (UQAM)（蒙特利尔大学）
- Département de finance, École des sciences de la gestion, Université du Québec à Montréal (UQAM)（蒙特利尔大学）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
针对扩散盈余模型，研究股息支付与资本注入的联合优化，提出折射-反射策略，并证明最优策略存在Lokka-Zervos型二分法。
中文摘要 AI 辅助
我们研究了一个由一般扩散过程驱动的盈余过程中的股息支付与资本注入联合优化问题。假设股息支付在时间上是绝对连续的，股息率受当前盈余的非负凹函数约束；而资本注入则建模为一个一般非减过程。我们首先分析了一个辅助的纾困问题，在该问题中，资本注入被要求始终保持盈余非负。在漂移项的凹性假设下，我们证明了相应的值函数是凹的，并且是对应Hamilton-Jacobi-Bellman (HJB)方程的经典解。我们进一步将最优策略刻画为折射-反射策略：盈余通过资本注入在零处反射，而当盈余超过最优阈值时，以最大允许速率支付股息。我们的主要贡献在于证明了该一般优化问题表现出Lokka-Zervos型二分法。更确切地说，最优策略要么是无注入的股息折射策略（此时发生破产），要么是折射-反射的股息-注入策略。这一最优注入决策通过原点处辅助值函数的简单比较来刻画。
英文摘要
We investigate a joint optimization problem of dividend payments and capital injections for a surplus process driven by a general diffusion. Dividend payments are assumed to be absolutely continuous in time, with the dividend rate bounded by a nonnegative concave function of the current surplus; while capital injections are modelled by a general nondecreasing process. We first analyze an auxiliary bail-out problem in which capital injections are required to keep the surplus nonnegative at all times. Under a concavity assumption on the drift, we prove that the associated value function is concave and is a classical solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We further characterize an optimal policy as a refraction-reflection strategy: the surplus is reflected at zero by capital injections, while dividends are paid at the maximal admissible rate whenever the surplus exceeds an optimal threshold. Our main contribution establishes that the general optimization problem exhibits a Lokka-Zervos type dichotomy. More precisely, an optimal policy is either a dividend refraction strategy without injections, in which ruin occurs, or a refraction-reflection dividend-injection strategy. This optimal injection decision is characterized through a simple comparison of the auxiliary value functions at the origin.
