CAREER: Equilibria and Stability in Financial Markets

从定量的角度理解金融市场的目标是重要的，不仅因为它的知识价值，而且因为它在风险管理或监管中的作用。均衡方法将久经考验的经济见解与现代数学工具相结合，以实现这一目标。这个项目将使用均衡方法来进一步掌握所谓的不完全市场，即，这些市场就像现实世界的市场一样，降低风险的能力有限。研究者将研究一些问题，这些问题围绕着不完全连续时间金融市场中随机均衡的概念。拟议研究的目标是双重的。在概念层面上，它的目的是提供一个通用的方法，从市场原语的连续时间资产价格模型的规格，并建立一个新的建模框架，分析和更好地理解不完整的金融市场及其均衡动态。引入了一个新的概念，完整性约束，以捕捉卓有成效的想法，市场的不完全性可以解释为一个外生的约束。 在技术层面上，它基于现有的和新的随机、凸和函数分析以及基于偏微分方程的方法，为上述框架的数学分析提供了工具。这些工具，然后，用于建立平衡的存在性，并研究其性质。 数学的核心是研究作为建模输入函数的随机控制问题的最优解的稳定性概念。所得到的稳定性结果可以解释的总需求函数，并用于授予存在的均衡市场的帮助下，适当的（古典和新的）不动点定理。

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.