Systems of Backward Stochastic Differential Equations and Applications in Stochastic Financial Equilibrium Theory

University Of Texas At Austin, Austin TX

Investigators

Abstract

Why do stock-price graphs look the way they do? Why do market crashes and rallies occur? How much do asset prices depend on our risk appetites, and how much on the inherent randomness of the world around us? These questions can be answered by analyzing various mathematical models based on the fundamental idea that prices adjust to equalize supply with demand. This analysis is often very hard, and requires considerable analytical effort, but is nevertheless within the reach of contemporary mathematics and its tools. The societal benefits of a better understanding of the structure behind asset-price movements are numerous; for instance, it will help us regulate the markets more efficiently and plan better for potential future financial instabilities. The principal investigator and his students and collaborators will study several separate, but ultimately related, clusters of problems at the intersection of probability, mathematical finance and stochastic-control theory. One area of concentration is the class of equations, known as Backward Stochastic Differential Equations (BSDEs), which describe evolutions of random processes in many different contexts. The PI and his collaborators will focus on the existence and uniqueness theory for systems of nonlinear fully-coupled BSDEs. While their interest in this subject originates in its pivotal role in the financial equilibrium theory, there are numerous other applications in finance, economics, game theory, optimal stochastic control, and other fields. The second cluster of problems centers around the so-called mean-field limits in the context of equilibrium theory. The main goal here is to provide a meaningful simplification of the equilibrium problem when the market consists of a large number of small economic agents. Attention will be paid to the limiting behavior of the systems of BSDEs which appear in incomplete-market equilibrium models with exponential investors. 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.