New perspectives in contract theory: Optimal incentives for interacting agents in a common random environment

连续时间委托代理问题为研究代理人之间的最优激励提供了一个相关的数学框架，特别是在信息不对称的情况下。在Holmström和Milgrom(1987)的开创性模型中，委托人(她)不完全了解代理人(他)对代表项目随时间的价值的随机过程的操作。为了激励代理人按照她的最佳利益行事，她可以向他提供一份合同，即根据项目价值确定的终止金。这些问题从根本上与最优激励的设计有关，因此存在于各种情况下，不仅包括经济，而且包括政治、金融等。尽管这一理论已被扩展到允许委托人与许多代理人签约，但代理人可能受到共同危险和风险影响的可能性目前大多被忽视。本研究着眼于发展委托-代理问题，以包含代理人可能生活在共同的不确定环境中，并且可能相互作用，但也与此环境相互作用的事实。这一研究主题受到几个具体应用的推动，在这些应用中，在寻找最佳行动或激励措施实施时，不能忽视这种共同的不确定环境，例如，优化电力生产和消费、监管金融和系统性风险、或设计最佳和可持续的保险单。这些主题是当前国家和国际经济和社会挑战的核心。因此，以定量的方式研究它们可以帮助为公共政策提供信息，从而有助于实现具有社会意义的成果。这项研究还将有一个重要的指导方向，涉及运筹学和金融工程研究生项目的研究生，他们将得到部分奖励资金的支持。考虑到共同的随机环境会引发广泛的额外数学困难，这些困难最近才得到解决，尽管只适用于纯平均场游戏。为了解决这类一般性问题，将进一步发展技术成果，特别是关于二阶倒向随机微分方程(2BSDE)的技术成果，该方程通常用于确定委托-代理问题中的最优合同形式。为了考虑具有Nash或平均场相互作用的一般多智能体问题，有必要建立2BSDE的广义概念，即多维或平均场2BSDE。此外，即使在经典框架中，考虑多因素或平均场环境中的共同跳跃的想法从未被研究过，尽管它与气候灾害等模拟集体事故有关，并将涉及具有跳跃的(多维或平均场)2BSDE的研究。最后，在保证现实合同执行的前提下，委托人对合同的约束问题可以转化为随机目标问题。这些理论发展将大大提高2BSDE领域和更广泛的随机控制领域的知识，并将允许研究各种应用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Continuous-time principal-agent problems offer a relevant mathematical framework for the study of optimal incentives between agents, especially with information asymmetry. In the seminal model by Holmström and Milgrom (1987), a principal (she) is imperfectly informed about the actions of an agent (he) on a random process representing the value of a project over time. To incentivize the agent to act in her best interest, she can offer him a contract, namely a terminal payment indexed on the value of the project. These problems are fundamentally related to the design of optimal incentives and are therefore present in a wide variety of situations, including not only economics but also politics, finance, etc. Although this theory has been extended to allow the principal to contract with many agents, the possibility that agents may be impacted by common hazards and risks is currently mostly neglected. This research focuses on the development of principal-agent problems to incorporate the fact that the agents may live in a common uncertain environment and may interact with each other but also with this environment. This research theme is motivated by several concrete applications, where this common uncertain environment cannot be neglected when looking for the optimal actions or incentives to implement, e.g., optimization of electricity production and consumption, regulation of financial and systemic risks, or design of optimal and sustainable insurance policies. The themes are at the heart of current economical and societal challenges, both nationally and internationally. Studying them in a quantitative way can therefore help inform public policy, and thus contribute to the achievement of societally relevant outcomes. This research will also have an essential mentoring orientation, involving graduate students from the Operations Research & Financial Engineering Graduate Program, who will be partially supported by the funds awarded.Considering a common random environment induces a wide range of additional mathematical difficulties, which have only been recently addressed, albeit only for pure mean-field games. To address this type of general problem, technical results will be further developed, notably on second order backward stochastic differential equations (2BSDEs), which are typically used to determine the optimal form of contracts in principal-agent problems. To consider general multi-agent problems with Nash or mean-field interactions, it is necessary to develop generalized notions of 2BSDEs, namely multidimensional or mean-field 2BSDEs. Moreover, even in a classical framework, the idea of considering common jumps in a multi-agent or mean-field setting has never been investigated, despite its relevance to model collective accidents such as climatic hazards and will involve the study of (multidimensional or mean-field) 2BSDEs with jumps. Finally, with the idea of ensuring the implementation of realistic contracts, a principal’s problem with constraints on the contract can be reformulated as a stochastic target problem. These theoretical developments will considerably advance knowledge in the field of 2BSDE and more broadly of stochastic control and will allow the study of the various applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.