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

这个研究项目包含三个独立但相关的调查途径。这三个问题都是受到金融和经济学实际问题的启发或与之相关的。第一个项目侧重于一类模型，称为Kyle模型，它描述了金融市场参与者之间如何交换信息。例如，这些模型可以用来检测内幕交易。第二个项目的重点是一类方程，称为倒向随机微分方程，以更好地理解代理在广泛竞争环境中的战略行为。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，不仅是通过改善PI自己机构的咨询和教学，而且还通过在国内和国际上传播所产生的研究成果和知识。关于Kyle建模的第一个项目侧重于PI及其合作者最近的工作，其中研究了随机波动的“噪声交易”。PI计划在各个方向扩展这些结果，例如引入几个资产，研究与一个有趣的优化问题的关系，以及对随机过程的相关非标准分解问题的调查。第二个项目将解决与非线性、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几个问题。特别是，PI计划继续这项研究，并使用基于过去所谓的“次鞅”表征以及具有bmo系数的线性系统的结果的方法来关注非马尔可夫情况。第三个项目将研究一类被称为Hawkes过程的点过程在“几乎不稳定”状态下的极限理论。PI和他的学生将对自激具有“长尾”的课程特别感兴趣。预计这类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。

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.