In the first part of this project, the investigator and his colleagues seek to use methods from stochastic analysis to provide a comprehensive study of price formation in financial markets such as the stock and fixed-income markets. The specific goal is to understand how idiosyncratic risks affect equilibrium price formation. Currently, there are very few results available that can answer such questions and this part of the project seeks to provide tractable models that can be used to obtain approximate answers. In the second part, the investigator and his colleagues seek to develop optimization tools that can deal with a large class of stochastic control problems often encountered in mathematical finance. These problems exhibit unexpected discontinuities that prevent the standard mathematical tools from being applicable.
From a rigorous mathematical perspective, the investigator first seeks to establish the existence of incomplete equilibria in continuous-time models governed by Brownian motions. Such models are well-known for being notoriously intractable. Therefore, the investigator seeks to provide tractable approximation tools that can be used as surrogates for the general models. Secondly, the investigator and his colleagues seek to provide a partial differential equation characterization of the problem of optimal investment with unspanned endowment. This control problem turns out to have a discontinuous value function (a facelift or boundary layer), which prevents the use of standard partial differential equation techniques. Finally, the investigator and his colleagues seek to develop tools that can detect up front such discontinuities for a general class of stochastic control problems.
