CAREER: Equilibria and Stability in Financial Markets

从定量的角度理解金融市场的目标不仅因为它的智力价值，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学见解与现代数学工具相结合，以实现这一目标。该项目将使用均衡方法来进一步掌握所谓的不完全市场，即市场-就像现实世界的市场一样-具有有限的风险缓解能力。研究者将研究一些围绕不完全连续时间金融市场随机均衡概念的问题。拟议研究的目标是双重的。在概念层面上，它旨在为从市场原语出发的连续时间资产价格模型的规范提供一种通用方法，并为分析和更好地理解不完全金融市场及其均衡动态建立一个新的建模框架。为了抓住市场不完全性可以被解释为外生强加的约束这一富有成效的思想，引入了完备性约束的新概念。在技术层面上，它基于现有的和新的随机分析、凸分析和泛函分析以及基于pde的方法，为上述框架的数学分析提供了工具。然后，这些工具被用来建立平衡的存在性并研究它们的性质。数学核心是研究作为建模输入函数的随机控制问题最优解的稳定性概念。得到的稳定性结果可以用总需求函数来解释，并利用适当的（经典的和新的）不动点定理来证明均衡市场的存在性。

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.