Hawkes processes are studied and used in a wide range of disciplines: mathematics, social sciences, and earthquake modelling, to name a few. This book presents a selective coverage of the core and recent topics in the broad field of Hawkes processes. It consists of three parts. Parts I and II summarise and provide an overview of core theory (including key simulation methods) and inference methods, complemented by a selection of recent research developments and applications. Part III is devoted to case studies in seismology and finance that connect the core theory and inference methods to practical scenarios.
This book is designed primarily for applied probabilists, statisticians, and machine learners. However, the mathematical prerequisites have been kept to a minimum so that the content will also be of interest to undergraduates in advanced mathematics and statistics, as well as machine learning practitioners. Knowledge of matrix theory with basics of probability theory, including Poisson processes, is considered a prerequisite. Colour-blind-friendly illustrations are included.
Author(s): Patrick J. Laub, Young Lee, Thomas Taimre
Publisher: Springer
Year: 2022
Language: English
Pages: 147
City: Cham
Preface
Contents
Notation
1 Introduction
Part I Basic Theory
2 Background
2.1 Counting and Point Processes
2.2 Poisson Processes
2.3 Conditional Intensity Functions
2.4 Compensators
3 Hawkes Process Essentials
3.1 The Hawkes Process
3.2 Hawkes Conditional Intensity Function
3.3 Immigration–Birth Representation
3.4 Markov Property for Exponential Decay
3.5 Covariance and Power Spectral Densities
3.6 Generalisations
4 Simulation Methods
4.1 Transformation Methods
4.2 Exact Simulation with Exponential Decay
4.3 Ogata's Modified Thinning Algorithm
4.4 Superposition of Poisson Processes
4.5 Mutually Exciting Hawkes Processes
Part II Inference
5 Maximum Likelihood Estimation
5.1 Likelihood Function
5.2 Simplifications for Exponential Decay
5.3 Likelihood for Mutually Exciting Hawkes Processes
5.4 Discussion
6 EM Algorithm
6.1 EM Algorithm for Hawkes Processes
6.1.1 Complete Data Log-Likelihood
6.1.2 The E Step
6.1.3 The M Step
6.1.4 The Algorithm
6.2 The Quasi-EM Algorithm
6.3 A Worked Example
6.3.1 The EM Algorithm
6.3.2 Quasi-EM Algorithm
6.3.3 Results
7 Moment Matching and Interval Censored Inference
7.1 The Generalised Method of Moments
7.1.1 Method of Moments
7.1.2 Generalised Method of Moments
7.2 Application to Hawkes Processes
7.2.1 Moments Involving λ(t)
7.2.2 Moments Involving N(t) and λ(t)
7.3 Numerical Results and Discussion
7.3.1 GMM for Hawkes Model
7.4 Inference for Generalised Hawkes
8 Bayesian Methods
8.1 A Primer on Bayesian Inference and MCMC
8.2 Bayesian Inference for Random Hawkes Processes
8.2.1 The Likelihood
8.2.2 The Priors
8.2.3 The Posteriors
8.2.4 Markov Chain Monte Carlo
8.2.5 The Proposals
8.2.6 The Acceptance Ratios
8.3 Experiments
9 Goodness of Fit
9.1 Transformation to a Poisson Process
9.2 Tests for Poisson Process
9.2.1 Basic Tests
9.2.2 Test for Independence
9.2.3 Lewis Test
9.2.4 Brownian Motion Approximation Test
9.3 Mutually Exciting Hawkes Processes
9.4 Exponentially Decaying Kernels
Part III Case Studies
10 Code Preliminaries
10.1 Intensity Functions and Compensators
10.2 Log-Likelihoods and MLE
10.3 Simulation
10.4 Fitting
10.5 Mutually Exciting Hawkes Processes
11 Seismology
11.1 Data Preparation and Exploration
11.2 Poisson Process
11.3 Hawkes Process with Exponential Decay
11.4 Hawkes Process with Power Law Decay
11.5 Discussion
12 Finance
12.1 Data Preparation and Exploration
12.2 Independent Poisson Processes
12.3 Independent Hawkes Processes
12.4 Mutually Exciting Hawkes Processes
12.5 Discussion and Literature Review
12.5.1 Financial Contagion
12.5.2 Mid-Price Changes and High-Frequency Trading
A Supplementary Material
A.1 Preliminary Background Concepts
A.2 Additional Proof Details
A.2.1 Supplementary to Theorem 3.2 (Part I)
A.2.2 Supplementary to Theorem 3.2 (Part II)
References
