Self-Exciting Point Processes and Their Applications

在数学中，点过程是纯跳跃随机过程，这意味着它们是在随机时间跳跃的时间演化，并且可能具有随机跳跃大小。 这些过程对于在社会学、生物学、犯罪学、地震学、金融学和许多其他领域的研究中出现的复杂系统建模是有用的。最标准的点过程是具有独立时间增量的泊松过程，这意味着未来的跳跃与过去发生的事情无关。然而，在现实世界的数据中，人们并不经常观察到独立的时间增量。例如，正如在2008年金融危机期间所看到的那样，信用违约具有传染效应-一家公司的违约可能引发更多公司违约。在社交网络中，用户重新共享由他们的社交联系发布的内容的可能性可以跨系统级联。在地震学中，地震发生后，该地区经常发生余震。在犯罪学中，与帮派有关的暴力具有这样的性质，即一个帮派的谋杀或枪击往往会引起另一个帮派的报复。所有这些例子都有一个共同点，即自激特性：一个事件可以触发更多的事件。自激点过程，本研究项目的主题，构成了一类点过程，可以描述这样的现象。尽管它们在应用中很重要，但许多关键的核心事实仍然未知。本计画将探讨自激点过程的理论及其应用。它旨在促进理解复杂和大型随机系统如何自我和相互激发，相互作用，集群和影响传染。从这一框架中得出的结果将被应用于更好地理解真实的世界中复杂系统产生的大数据集。更具体地说，这个项目研究了一类自激点过程，重点是霍克斯过程及其扩展，包括非线性霍克斯过程。到目前为止，该模型的极限定理仅限于大时间渐近性，除了多元非线性情况外，它们已经得到了很好的研究，这将在本研究中进行研究。研究人员打算了解固定时间间隔上的大渐近性，这在许多应用中非常有用。这些大的渐近将不同于泊松过程和相变是预期的。研究人员还将研究多维Hawkes过程的平均场极限的波动和大偏差，当维度很大时，这在神经网络，金融网络等应用中很有用。该项目还将探索空间霍克斯过程，时空霍克斯过程和其他类型的自激点过程，这些过程已被建议用于建模，但仍然缺乏一些关键的理论结果。研究人员还旨在研究自激点过程的应用，包括理论应用到自激理论和将这些模型拟合到真实的世界大数据集，这需要更好地理解自激点过程的模拟和校准的一些理论方面。

In mathematics, point processes are pure-jump stochastic processes, which means they are time evolutions that jump at random times and may have random jump sizes. These processes are useful to model the complex systems that arise in the study of sociology, biology, criminology, seismology, finance, and many other fields. The most standard point process is the Poisson process that has independent time increments, which means that future jumps are independent of what has occurred in the past. However, one does not often observe independent time increments in real-world data. For example, as seen during the 2008 financial crisis, credit defaults have a contagion effect -- a default of one company can trigger more companies to default. In social networks, the possibility for users to re-share the content posted by their social connections may cascade across the system. In seismology, after an earthquake hits, the area often experiences aftershocks. In criminology, gang-related violence has the property that a murder or shooting by one gang often provokes retaliation by another gang. All these examples have in common the self-exciting property: an event can trigger more events to come. Self-exciting point processes, the subject of this research project, constitute a class of point processes that can describe such phenomena. Despite their importance in applications, many key central facts are still unknown. This project will investigate the theoretical aspects of self-exciting point processes, as well as their applications. It aims to advance understanding of how complex and large stochastic systems self and mutually excite, interact, cluster, and effect contagion. The results derived from this framework will be applied to better understand the big data sets arising from complex systems in the real world. More specifically, this project studies a class of self-exciting point processes, with a focus on the Hawkes process and its extensions, including the nonlinear Hawkes process. To date, the limit theorems for this model are restricted to large time asymptotics, and they have been well studied except the multivariate nonlinear case, which will be investigated in this research. The investigator intends to understand the large asymptotics on a fixed time interval, which will be very useful in the context of many applications. These large asymptotics will differ from the Poisson process and phase transitions are anticipated. The investigator will also study the fluctuations and large deviations of the mean-field limit of a multivariate Hawkes process when the dimension is large, which is useful in applications to neural networks, financial networks, and others. The project will also explore the spatial Hawkes processes, the space-time Hawkes processes, and other types of self-exciting point processes that have been suggested for modeling but still lack some key theoretical results. The investigator also aims to study applications of the self-exciting point processes, including theoretical applications to queueing theory and to fitting these models to real world big data sets, which requires a better understanding of some theoretical aspects of the simulations and calibrations of self-exciting point processes.