[Paper Review] Hawkes Processes

Patrick J. Laub, Thomas Taimre+1 more|arXiv (Cornell University)|Jul 10, 2015

Point processes and geometric inequalitiesMathematics3 references22 citations

TL;DR

This survey introduces Hawkes processes—a class of self-exciting point processes that model events triggering future events—offering a comprehensive, accessible overview of their theoretical foundations, historical development, and applications in finance, seismology, and beyond. The key contribution is a unified, approachable synthesis of core concepts, equations, and practical implementations for researchers new to the field.

ABSTRACT

Hawkes processes are a particularly interesting class of stochastic process that have been applied in diverse areas, from earthquake modelling to financial analysis. They are point processes whose defining characteristic is that they 'self-excite', meaning that each arrival increases the rate of future arrivals for some period of time. Hawkes processes are well established, particularly within the financial literature, yet many of the treatments are inaccessible to one not acquainted with the topic. This survey provides background, introduces the field and historical developments, and touches upon all major aspects of Hawkes processes.

Motivation & Objective

• To provide a clear, accessible introduction to Hawkes processes for researchers unfamiliar with the topic.
• To trace the historical development and foundational concepts of self-exciting point processes.
• To consolidate major theoretical and applied aspects of Hawkes processes across diverse domains.
• To bridge the gap between advanced mathematical treatments and practical research applications.
• To serve as a foundational reference for researchers entering the field of point process modeling.

Proposed method

• The paper employs a narrative, expository approach to explain the stochastic foundations of Hawkes processes.
• It introduces the conditional intensity function, the core mathematical mechanism driving self-excitation.
• Key equations include the stochastic integral representation of the intensity process and the branching process representation.
• The survey contextualizes the process within point process theory, emphasizing its self-exciting nature.
• It discusses estimation techniques, such as likelihood-based inference, and simulation methods.
• Applications are illustrated through examples in finance and earthquake modeling to demonstrate practical relevance.

Experimental results

Research questions

• RQ1 How do Hawkes processes model self-excitation in event sequences?
• RQ2 What are the fundamental mathematical structures and equations underpinning Hawkes processes?
• RQ3 How have Hawkes processes evolved historically and in application across different scientific domains?
• RQ4 What are the key challenges in estimating and simulating Hawkes processes?
• RQ5 How can Hawkes processes be effectively applied in real-world scenarios such as financial risk or seismic activity?

Key findings

• Hawkes processes are defined by a conditional intensity function that increases after each event, capturing self-excitation.
• The process can be represented as a superposition of immigrant events and their descendants, forming a branching structure.
• The likelihood function for observed event sequences is derived using the stochastic intensity, enabling statistical inference.
• The survey highlights the widespread applicability of Hawkes processes in modeling clustering phenomena in diverse systems.
• It emphasizes the importance of the excitation kernel in determining the temporal dependence structure of event arrivals.
• The paper establishes that Hawkes processes provide a flexible and mathematically tractable framework for modeling endogenous event clustering.