Interest Rates and Transaction Costs in Incomplete Equilibria

金融模型的典型起点是将资产价格作为给定的，并对其进行操纵以确定结果。另一种研究较少的途径是问，是什么让资产定价模型好，这些好模型从何而来？均衡理论研究的是一个经济体的总供给满足其总需求的基本水平上资产定价模型的形成。该奖项将研究两个在不完全环境下设定的总体均衡项目，其中并非所有风险都可以通过交易捕获。这些项目使用数学工具来描述基于均衡的“良好”模型，涉及(i)利率的形成，以及（ii）存在比例交易成本的股票价格的形成。利率和交易成本都与金融市场高度相关，但目前在不完全均衡环境下理解得很少。该奖项将涉及合作以及研究生的参与和培训。该奖项将研究数学金融中涉及随机系统均衡的两个相关问题。第一个目标是证明具有随机利率和比例交易成本的不完全市场模型中有限代理Radner均衡的存在性。在获得存在性后，研究者将分析不完备性对内生决定资产价格的影响。在数学上，分析利用了完全耦合的二次倒向随机微分方程（BSDE）系统，该系统在交易成本的情况下是双重反射的。这类系统的存在性结果是非建设性的，使得不完备效应的分析变得复杂。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

A typical starting point for a financial model takes an asset price as given and manipulates it to determine results. A less studied path asks what makes an asset pricing model good, and where do such good models come from? Equilibrium theory studies the formation of asset pricing models at a fundamental level where an economy's total supply meets its total demand. This award will study two overarching equilibrium projects set in incomplete settings, where not all risk can be captured by trading. The projects use mathematical tools that are now within reach to describe equilibrium-based "good" models involving (i) the formation of interest rates, and (ii) the formation of stock prices in the presence of proportional transaction costs. Both interest rates and transaction costs are highly relevant to financial markets but currently poorly understood in incomplete equilibrium settings. The award will involve collaborations and the involvement and training of graduate students. This award will study two related problems involving equilibrium of stochastic systems in mathematical finance. The first goal is to prove the existence of a finite-agent Radner equilibrium in incomplete market models with stochastic interest rates and proportional transaction costs. After obtaining existence, the investigator will analyze the effects of incompleteness on the endogenously determined asset prices. Mathematically, the analysis draws from systems of fully coupled quadratic backward stochastic differential equations (BSDE), which are doubly reflective in the case of transaction costs. Existence results for such BSDE systems are nonconstructive, which complicates the analysis of incompleteness effects.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.