The investigator studies two problems related to behavior of financial markets. In the first he studies equilibrium in incomplete markets. The theory of market equilibrium is important because it provides a framework for understanding how prices develop in a market. Roughly speaking, a market is complete if every agent can trade with every other agent about every possible future state. The theory for complete markets has been worked out reasonably well, but the equilibrium of incomplete markets is open. In the second, he examines stochastic control problems that arise in considering how to make optimal investments with a random endowment. Formation of financial markets is a complex process, but ultimately reliant on fundamental economic principles. This reliance is what makes it amenable to mathematical analysis and quantitative study. The benefits of better understanding of how real financial markets come to exist, and how their salient features emerge from the basic building blocks, are multiple. They range from enhancing our ability to regulate both existing and nascent markets to building tools for prevention of future financial instabilities and crises in the US and worldwide. Graduate students are included in the project.
The investigator and his students study two separate, but related, clusters of problems in mathematical finance and stochastic-control theory. The first one aims to elucidate the rational agents' optimal response to diverse market structures and the formation of incomplete financial markets. In addition to the classical, convex- and functional-analytic tools, the investigator's approach rests on stochastic analysis and the theory of backward stochastic differential equations, as well as their relationship with systems of partial differential equations, touching upon topics in Riemannian geometry and non-cooperative game theory. The second research focus -- centered around a novel class of stochastic control problems termed "weakly constrained" -- is modeled after one of the fundamental questions in mathematical finance, namely the optimal investment problem with a random endowment. These weakly constrained problems occur naturally and possess intriguing mathematical features, but resist analysis using standard approaches.
