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.
