I Introduction
1 Prologue
Why Is the Martingale So Useful?
Martingale as a tool to analyze time series data in real time
Martingale as a tool to deal with censored data correctly
Invitation to Statistical Modelling with Semimartingales
From non-linear regression to diffusion process model
Cox’s regression model as a semimartingale
2 Preliminaries
Remarks on Limit Operations in Measure Theory
Limit operations for monotone sequence of measurable sets
Limit theorems for Lebesgue integrals
Conditional Expectation
Understanding the definition of conditional expectation
Properties of conditional expectation
Stochastic Convergence
3 A Short Introduction to Statistics of Stochastic Processes
The "Core" of Statistics
Two illustrations
Filtration, martingale
Motivation to Study Stochastic Integrals
Intensity processes of counting processes
Itˆo integrals and diffusion processes
Square-Integrable Martingales
Predictable quadratic variations
Stochastic integrals
Introduction to CLT for square-integrable martingales
Asymptotic Normality of MLEs in Stochastic Process Models
Counting process models
Diffusion process models
Summary of the approach
Examples
Examples of counting process models
Examples of diffusion process models
II A User’s Guide to Martingale Methods
4 Discrete-Time Martingales
Basic Definitions, Prototype for Stochastic Integrals
Stopping Times, Optional Sampling Theorem
Inequalities for 1-Dimensional Martingales
Lenglart’s inequality and its corollaries
Bernstein’s inequality
Burkholder’s inequalities
5 Continuous-Time Martingales
Basic Definitions, Fundamental Facts
Discre-Time Stochastic Processes in Continuous-Time
φ(M) Is a Submartingale
"Predictable" and "Finite-Variation"
Predictable and optional processes
Processes with finite-variation
A role of the two properties
Stopping Times, First Hitting Times
Localizing Procedure
Integrability of Martigales, Optional Sampling Theorem
Doob-Meyer Decomposition Theorem
Doob’s inequality
Doob-Meyer decomposition theorem
Predictable Quadratic Co-Variations
Decompositions of Local Martingales
6 Tools of Semimartingales
Semimartingales
Stochastic Integrals
Starting point of constructing stochastic integrals
Stochastic integral w.r.t. locally square-integrable martingale
Stochastic integral w.r.t. semimartingale
Formula for the Integration by Parts
Itˆo’s Formula
Likelihood Ratio Processes
Likelihood ratio process and martingale
Girsanov’s theorem
Example: Diffusion processes
Example: Counting processes
Inequalities for 1-Dimensional Martingales
Lenglart’s inequality and its corollaries
Bernstein’s inequality
Burkholder-Davis-Gundy’s inequalities
III Asymptotic Statistics with Martingale Methods
7 Tools for Asymptotic Statistics
Martingale Central Limit Theorems
Discrete-time martingales
Continuous local martingales
Stochastic integrals w.r.t. counting processes
Local martingales
Functional Martingale Central Limit Theorems
Preliminaries
The functional CLT for local martingales
Special cases
Uniform Convergence of Random Fields
Uniform law of large numbers for ergodic random fields
Uniform convergence of smooth random fields
Tools for Discrete Sampling of Diffusion Processe
8. Parametric Z-Estimators
Illustrations with MLEs in I.I.D. Models
Intuitive arguments for consistency of MLEs
Intuitive arguments for asymptotic normality of MLEs
General Theory for Z-estimators
Consistency of Z-estimators, I
Asymptotic representation of Z-estimators, I
Examples, I-1 (Fundamental Models)
Rigorous arguments for MLEs in i.i.d. models
MLEs in Markov chain models
Interim Summary for Approach Overview
Consistency
Asymptotic normality
Examples, I-2 (Advanced Topics)
Method of moment estimatorsQuasi-likelihood for drifts in ergodic diffusion models
Quasi-likelihood for volatilities in ergodic diffusion modelsPartial-likelihood for Cox’s regression models
More General Theory for Z-estimators
Consistency of Z-estimators, II
Asymptotic representation of Z-estimators, II
Example, II (More Advanced Topic: Different Rates of Convergence)Quasi-likelihood for ergodic diffusion models
9 Optimal Inference in Finite-Dimensional LAN Models
Local Asymptotic Normality
Asymptotic Efficiency
How to Apply
10 Z-Process Method for Change Point Problems
Illustrations with Independent Random Sequences
Z-Process Method: General Theorem
Examples
Rigorous arguments for independent random sequences
Markov chain models
Final exercises: three models of ergodic diffusions
A Appendices
A1 Supplements
A1.1 A Stochastic Maximal Inequality and Its Applications
A1.1.1 Continuous-time case
A1.1.2 Discrete-time case
A1.2 Supplementary Tools for the Main Parts
A2 Notes
A3 Solutions/Hints to Exercises
Biography
Yoichi Nishiyama is a professor in mathematical statistics and probability at the School of International Liberal Studies of Waseda University; he is also engaged in the education of master’s and doctoral students at the Department of Pure and Applied Mathematics at the same university. Prior to his assignment to Waseda University, he worked at the Institute of Statistical Mathematics, Tokyo, from 1994 to 2015. He was the Editor-in-Chief of Journal of the Japan Statistical Society and a Co-Editor of Annals of the Institute of Statistical Mathematics and he received the JSS Ogawa Award from the Japan Statistical Society in 2009.
"This book is expected to be an excellent reference for researchers who need to perform statistical analysis based on the martingale theory. It is very rare to find a book that systematically and rigorously summarizes martingale theory from the point of view of its application to statistics. This textbook harmonically organizes the mathematical facts related to martingales and their statistical applications, and by doing so, it helps researchers to establish a theoretically concrete foundation. Therefore, this book can be evaluated as an excellent textbook where mathematics and statistics meet together."
-Insuk Seo, in Journal of the American Statistical Association, November 2023"The martingale theory is an important topic in probability theory and related tools have been widely applied in statistical analysis, such as financial data or survival analysis. ...This book well summarizes useful tools in martingale and provides rigorous theorems. ... In summary, this book is a nice reference because of rich and comprehensive materials in martingale
theory. This book is suitable to researchers who are working on related research topics."
- Li-Pang Chen, in Journal of the Royal Statistical Society: Series A, April 2022
