The goal of understanding the financial markets from the quantitative perspective is important not only for its intellectual merit, but also for its role in risk management or regulation. The equilibrium approach combines the time-tested economic insights with modern mathematical tools to achieve this goal. This project will use the equilibrium approach to further our grasp of so-called incomplete markets, i.e., the markets which--just like the real-world markets--have a limited ability to mitigate risk.
The investigator will study a number of questions centered around the notion of a stochastic equilibrium in incomplete continuous-time financial markets. The goals of the proposed research are two-fold. On the conceptual level, it aims to provide a general methodology for the specification of continuous-time asset-price models from the market primitives and to establish a new modelling framework for the analysis and better understanding of incomplete financial markets and their equilibrium dynamics. A novel concept of completeness constraint is introduced in order to capture the fruitful idea that market incompleteness can be interpreted as an exogenously-imposed constraint. On the technical level, it furnishes tools for the mathematical analysis of the framework described above, based on the existing and new stochastic-, convex- and functional-analytic and PDE-based methods. These tools are, then, used to establish the existence of equilibria and to study their properties. The mathematical core is the study of the notion of stability for the optimal solutions of stochastic control problems as functions of the modelling inputs. The obtained stability results can be interpreted in terms of the aggregate demand functions and used to grant existence of equilibrium markets with the help of the appropriate (classical and new) fixed-point theorems.
