Dynamic Optimization in Management

这项研究计划的目标是开发数学模型和有效的算法来解决实际决策问题，这些问题本质上是动态的，涉及不确定性。将考虑两个具体的应用领域：金融市场的投资决策，以及环境和自然资源的管理。虽然这些应用领域非常不同，但它们都有一个共同的方法论方法，即多维框架下的随机动态优化。在这两种情况下，解决现实问题都需要开发有效的数值方法。因此，除了解决这些具体问题外，这项研究计划的一个更广泛的目标是开发大规模随机动态问题的数值解的创新方法，并为在不确定情况下的决策领域的知识进步做出贡献。*对创新金融产品的评估几乎总是需要使用数字程序。此外，金融机构越来越关注风险管理和满足监管要求。在实践中，用于金融产品估值、风险评估和确定投资或对冲策略的算法需要实时使用，并应在几秒钟内提供准确的结果。这一线研究的目标是提高评估算法的效率(准确性、速度)，并开发可用于更现实的市场模型的新的通用方法。*环境和资源管理方面日益受到关注的问题与所谓的“公地悲剧”有关。获取共同资源(空气质量、生物多样性、公海捕捞、不可再生资源)会导致过度开发，并可能导致灾难性后果。防止这种情况发生的手段之一是在利益攸关方之间达成协议。在这种情况下，远见被定义为利益攸关方能够解释关键特征(环境破坏成本、学习、技术进步、签署国、经济发展)随时间演变的能力，而这种演变取决于代理人的联合战略。显然，动态和前瞻性是环境协议的设计、稳定性和可执行性的重要因素。这第二条研究的目标是研究协议如何受到动态和不确定性以及参与者的远见和战略前景的影响。*考虑的一般方法是多维状态空间的动态优化方法。解决方法将包括函数内插方案(值函数和/或最优策略)，以及降维技术，包括状态聚集和学习(向前)算法。**

The objective of this research program is to develop mathematical models and efficient algorithms to solve practical decision-making problems that are dynamic in nature and that involve uncertainty. Two specific areas of application will be considered: investment decisions in financial markets, and the management of environmental and natural resources. Although these application areas are very different, they share a common methodological approach, that is, stochastic dynamic optimization in a multidimensional framework. In both cases, the solution of realistic problems requires the development of efficient numerical methods. Thus, in addition to addressing these specific issues, a broader objective of this research program is to develop innovative methods for the numerical solution of large-scale stochastic dynamic problems, and to contribute to the advancement of knowledge in the field of decision-making under uncertainty. ***The evaluation of innovative financial products almost always requires the use of numerical procedures. Moreover, financial institutions are increasingly concerned with managing risk and satisfying regulatory requirements. In practice, algorithms for the valuation of financial products, the evaluation of risk, and the determination of investment or hedging strategies need to be used in real time and should provide accurate results in seconds. The objectives of this first line of research are to increase the efficiency (accuracy, speed) of evaluation algorithms, and to develop new general approaches that can be used for more realistic market models. ***A growing concern in environmental and resources management is related to the so-called "tragedy of the commons". Access to a common resource (air quality, biodiversity, high-seas fishing, non-renewable resources) leads to overexploitation, and possibly, to catastrophic consequences. One of the means employed to prevent this from happening is the establishment of agreements between stakeholders. In this context, foresight is defined as the ability of stakeholders to account for the evolution over time of key features (environmental damage costs, learning, technological progress, signatories, economic development), when this evolution depends on agents' joint strategies. Clearly, dynamics and foresight are important factors in the design, stability and implementability of environmental agreements. The objectives of this second line of research is to study how agreements are affected by dynamics and uncertainty, and by the foresight and strategic outlook of participants. ***The general methodology considered is that of dynamic optimization with multidimensional state spaces. Solution approaches will include functional interpolation schemes (value functions and/or optimal strategies), and dimension reduction techniques, including aggregation of states, and learning (forward) algorithms. **