Stochastic Games for Intergenerational Equity in Mathematical Finance

在社会财富和资源管理中，代际利益冲突是不可避免的。随着人口结构的变化达到关键时刻，这个问题现在比以往任何时候都更加紧迫。发达国家的劳动年龄总人口预计将持续减少，直至2050年，而退休人口将迅速增长。工作和退休(或年轻人和老年人)这两个群体之间的紧张关系几乎肯定会加剧，因为根据社会保障或许多其他养老金计划，前者的财富被用于支持后者。然而，确切的经济后果并不是那么明显。本项目旨在阐明不断变化的人口结构如何影响一个社会的财务规划。这将在两个不同的层面上进行：一个是从相互竞争的年轻人和老年人的角度来看，另一个是从社会规划者(如政府)的全景角度来看。研究生参与了这个项目。该项目的第一部分调查了年轻人和老年人之间的相互作用。它被描述为随机Ramsey模型中的两人非零和博弈，并对人口结构的动态进行了显式建模。我们的目标是了解这两个群体如何通过他们的消费和储蓄决策进行互动，以及不断变化的人口结构如何影响他们的决策和由此产生的福利。通过一个耦合的非Lipschitz系数的Hamilton-Jacobi-Bellman方程组，可以找到并分析这两个群体的纳什均衡策略。这就要求随机微分对策和粘性解技术有新的发展。该项目的第二部分解决了社会规划师的两难境地：从社会的角度来看，当前社会认为最优的财务规划战略在未来可能不是最优的，因为不断变化的人口结构不断改变整体社会偏好。这导致了一种新形式的时间不一致问题，具有非指数和依赖于时间的贴现函数，这将在当前和未来社会之间的个人内部博弈下处理。在该项目中，提出了一种新的时间不一致性方法--定点方法。它为构建时间一致的策略提供了一种新的机制，同时具有较少的数学技术细节的潜力。

Intergenerational conflicts of interest are inevitable to wealth and resource management in a society. This problem is more pressing now than ever as demographic change has reached a pivotal point. The combined working-age population in developed countries is projected to shrink continuously until 2050, while the retired population will grow rapidly. The tension between the two cohorts, working and retired (or, young and old), is almost certain to exacerbate, as the wealth of the former is used to support the latter, under Social Security or many other pension schemes. Yet, the precise economic consequences are not that apparent. This project aims to elucidate how an evolving demographic structure affects financial planning in a society. This will be done at two different levels: one from the eyes of the competing young and old cohorts, the other from the panoramic perspective of a social planner, such as a government. Graduate students are involved in the project. The first part of the project investigates the interplay between the young and the old cohorts. It is formulated as a two-player nonzero-sum game in a stochastic Ramsey model, with the dynamics of demographic structure explicitly modeled. The goal is to understand how the two cohorts interact through their consumption and saving decisions, and how an evolving demographic structure affects their decisions, and the resulting welfare. Nash equilibrium strategies of the two cohorts will be found and analyzed through a system of coupled Hamilton-Jacobi-Bellman equations with non-Lipschitz coefficients. This demands new developments in stochastic differential games and viscosity solution techniques. The second part of the project handles the social planner's dilemma: a financial planning strategy deemed optimal today by the current society may not be optimal from the view of the society in the future, as the evolving demographics continuously change the overall societal preference. This leads to a new form of time-inconsistent problems, with non-exponential and time-dependent discount functions, which will be treated under an intra-personal game between current and future societies. A new method for time-inconsistency, called the fixed-point approach, is developed in the project. It provides a new machinery for constructing time-consistent strategies, while possessing the potential of less mathematical technicalities.