Three Topics in Stochastic Analysis: Kyle's model, Systems of BSDEs and Superrough volatility

本研究项目包含三个独立但相关的调查途径。这三个问题都是由金融和经济学产生的实际问题所启发的，或者与之相关。第一个项目的重点是一类模型，称为凯尔模型，它描述了如何在金融市场的参与者之间交换信息。例如，这些模型可用于检测内幕交易。第二个项目的重点是一类方程，称为倒向随机微分方程，以更好地了解在广泛的竞争环境中的代理商的战略行为。最后，第三个项目旨在了解人们在金融市场波动中经常观察到的各种随机波动的动态和性质。最后，该项目将对教育和培训产生重大影响，不仅通过改进PI自己机构的咨询和教学，而且通过在国内和国际上传播所产生的研究成果和知识。第一个项目涉及凯尔建模，重点是PI及其合作者最近的工作，研究了随机波动的“噪声交易”。PI计划将这些结果扩展到各个方向，例如引入几种资产，研究与一个有趣的优化问题的关系，以及调查相关的非标准随机过程分解问题。第二个专题将解决几个与非线性、完全耦合、倒向随机微分方程系统解的存在性和唯一性有关的问题。特别是，PI计划继续这项研究，并专注于非马尔可夫的情况下，使用的方法的基础上，过去的结果，所谓的“下鞅”的特征，以及与BMO系数的线性系统。第三个项目将研究一类点过程的极限理论，称为霍克斯过程，在“几乎不稳定”的制度。PI和他的学生会对自激具有“长尾”的类特别感兴趣。预计这些过程的极限将属于一类新的随机场，与对数相关的高斯随机场有关。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This research project contains three separate but related avenues of inquiry. All three of them are either inspired by or are related to practical questions arising from finance and economics. The first project focuses on a class of models, known as Kyle models, which describe how information is exchanged among participants in a financial market. These models can be used, for example, to detect insider trading. The second project focuses on a class of equations, known as Backward Stochastic Differential Equations, to better understand strategic behavior of agents in a wide range of competitive environments. Finally, the third project aims to understand the dynamics and the nature of the kinds of random fluctuations one often observes in financial market volatility. Finally, the project will have a significant impact on education and training, not only through the improvement of advising and teaching at the PI's own institution, but also through national and international dissemination of the produced research findings and knowledge.The first project concerning Kyle modeling focuses on the recent work of the PI and his collaborators, where "noise trading" with stochastic volatility was studied. The PI plans to extend these results in various directions, such as the introduction of several assets, the study of the relationship with an intriguing optimization problem, and an investigation of a related nonstandard decomposition problem for stochastic processes. The second project will tackle several problems related to the existence and uniqueness of solutions for systems of nonlinear, fully-coupled, Backward Stochastic Differential Equations. In particular, the PI plans to continue this study and focus on the non-Markovian case using methods based on past results on the so-called "submartingale" characterization as well as on linear systems with BMO-coefficients. The third project will study the limiting theory for a class of point processes, known as Hawkes processes, in the "nearly unstable" regime. The PI and his students will be particularly interested in the class where the self-excitation has a "long tail". It is expected that the limits of such processes will belong to a new class of random fields, related to log-correlated Gaussian random fields.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.