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

随机分析的三个主题：凯尔模型、倒向随机微分方程系统和超粗糙波动性

• 批准号: 2307729
• 负责人: Gordan Zitkovic
• 金额: $ 44.13万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307729&HistoricalAwards=false
• 关键词: Three; Topics; Stochastic; Analysis; Kyle

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.

该研究项目包含三个独立但相关的探究途径。这三个都受到金融和经济学引起的实际问题的启发或有关。第一个项目的重点是一类称为凯尔模型的模型，这些模型描述了金融市场参与者之间如何交换信息。这些模型可用于检测内部交易。第二个项目着重于一类方程，称为后向随机微分方程，以更好地理解代理在广泛的竞争环境中的战略行为。最后，第三个项目旨在了解人们经常观察到金融市场波动的动态和性质。最后，该项目将对教育和培训产生重大影响，这不仅是通过PI自己的机构的建议和教学的改善，而且还通过对生产的研究发现和知识的国家和国际传播。有关Kyle建模的第一个项目，涉及PI及其合作者的最新工作，其中“与他的合作者进行了“与噪声交易”，其中与Stochostastic Volatisity进行了研究。 PI计划将这些结果扩展到各种方向，例如引入多个资产，与有趣的优化问题的关系以及对随机过程中相关的非标准分解问题的研究。第二个项目将解决与非线性，完全耦合，后退随机微分方程系统的存在和独特性有关的几个问题。特别是，PI计划继续使用基于过去的结果以及带有BMO-Coefficients的线性系统的结果来继续进行这项研究，并专注于非马克维亚案例。第三个项目将在“几乎不稳定”制度中研究一类称为霍克斯过程的一类过程的限制理论。 PI和他的学生将对自我激发具有“长尾巴”的班级特别感兴趣。预计此类过程的限制将属于与对数相关的高斯随机字段有关的新一类随机字段。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估的评估来获得支持的。