Mathematical Methods for the New Commodity & Environmental Markets

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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.