Mathematical Methods for the New Commodity & Environmental Markets

卡莫纳DMS-1211928 该项目的数学核心是研究具有大量玩家的随机微分游戏的近似平衡。 第一个挑战是启动概率的平均场博弈范式最初主张Lasry和狮子。 作为一个自然的延伸，研究者开发了McKean-Vlasov型随机微分方程的最优控制所需的理论和实践工具，尽管它的应用对理解大种群的行为很重要，但这个问题还没有被研究过。 研究人员开发了一种形式的庞特里亚金最大值原理适用于这些平均场动力学，推导出系统的平均场向前向后随机微分方程所产生的专门制作的伴随过程，并提供这些新类型的方程的存在性结果。 该项目的动机是过去几年中观察到的商品生产和价格变化的巨大社会影响。 投资于商品的机构数量的增加改变了这些商品的价格行为，以及它们之间的关系和与股票价格的关系。 此外，利用市场机制来限制温室气体排放，以期控制气候变化，也影响到其中一些市场。 该研究员专注于能源和排放市场激发的新理论问题，旨在开发工具，帮助风险管理者，监管机构和政策制定者应对这些新市场的挑战。

CarmonaDMS-1211928 The mathematical core of the project is the study of approximate equilibriums for stochastic differential games with a large number of players. The first challenge is to initiate the probabilistic approach to the mean-field game paradigm originally advocated by Lasry and Lions. As a natural extension, the investigator develops the theoretical and practical tools needed for the optimal control of stochastic differential equations of McKean-Vlasov type, a problem that has not been studied despite the importance of its applications to the understanding of the behavior of large populations. The investigator develops a form of the Pontryagin maximum principle appropriate for these mean-field dynamics, derives the systems of mean-field Forward-Backward Stochastic Differential Equations arising from specifically crafted adjoint processes, and provides existence results for these new types of equations. The project is motivated by the dramatic societal impacts of changes in the production and prices of commodities observed in the last few years. The increase in the number of institutions investing in commodities has changed the behavior of the prices of these commodities, and their relationships among themselves and with equity prices. Moreover, the use of market mechanisms to curb emissions of greenhouse gases in the hope of controlling climate change has also affected some of these markets. The investigator focuses on new theoretical problems motivated by the energy and emissions markets, and aims at the development of tools to help risk managers, regulators, and policy makers handle the challenges of these new markets.