线性二次优化下的稳定股息
Stable Dividends under Linear-Quadratic Optimization
AI总结：
本文针对风险业务的股息优化，提出在线性二次准则下为一般莱维过程推导最优仿射股息策略，通过HJB方程求解，实现明确要求的股息稳定性。
摘要：
风险业务股息的优化标准通常以未来股息的预期现值来形式化。该标准忽视了对于股息稳定性的潜在、明确需求。特别是在精算风险理论中，未来股息的最大化已被作为所谓的德·菲内蒂问题进行了深入研究。然而，那里的最优策略通常成为所谓的屏障策略。这些策略远非稳定，因此次优的仿射股息策略近年来受到关注。相比之下，在线性二次问题类别中，对稳定性的需求被明确强调。这些通常在与精算风险模型不同的扩散模型中被研究。我们通过在一个一般莱维过程下，依据线性二次标准推导最优仿射股息策略，弥合了这些思维模式之间的差距。我们通过哈密顿-雅可比-贝尔曼方程刻画价值函数，求解该方程，并将目标函数和最优控制与最大化未来股息预期现值的经典目标进行比较。由此，我们提供了一个框架，在该框架中，风险业务（如经典风险理论中）的股息稳定性被明确要求并明确实现。
英文摘要：
The optimization criterion for dividends from a risky business is most often formalized in terms of the expected present value of future dividends. That criterion disregards a potential, explicit demand for stability of dividends. In particular, within actuarial risk theory, maximization of future dividends have been intensively studied as the so-called de Finetti problem. However, there the optimal strategies typically become so-called barrier strategies. These are far from stable and suboptimal affine dividend strategies have therefore received attention recently. In contrast, in the class of linear-quadratic problems a demand for stability if explicitly stressed. These have most often been studied in diffusion models different from the actuarial risk models. We bridge the gap between these patterns of thinking by deriving optimal affine dividend strategies under a linear-quadratic criterion for a general Lévy process. We characterize the value function by the Hamilton-Jacobi-Bellman equation, solve it, and compare the objective and the optimal controls to the classical objective of maximizing expected present value of future dividends. Thereby we provide a framework within which stability of dividends from a risky business, as e.g. in classical risk theory, is explicitly demanded and explicitly obtained.
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