Semimartingale Theory and Stochastic Calculus

Sheng-Wu He, Jia-Gang Wang, Jia-an Yan, He/Wang/yan

• Publisher: Routledge
• Year: 2019
• Print ISBN:
  9780849377150
• Edition: 1
• eBook ISBN:
  9781351416955
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Semimartingale Theory and Stochastic Calculus presents a systematic and detailed account of the general theory of stochastic processes, the semimartingale theory, and related stochastic calculus. The book emphasizes stochastic integration for semimartingales, characteristics of semimartingales, predictable representation properties and weak convergence of semimartingales. It also includes a concise treatment of absolute continuity and singularity, contiguity, and entire separation of measures by semimartingale approach. Two basic types of processes frequently encountered in applied probability and statistics are highlighted: processes with independent increments and marked point processes encountered frequently in applied probability and statistics. Semimartingale Theory and Stochastic Calculus is a self-contained and comprehensive book that will be valuable for research mathematicians, statisticians, engineers, and students.

Chapter I

Preliminaries

Readers are assumed to be familiar with the fundamentals of measure theory and probability theory in Loève^[1] or Neveu^[1], such as measure extension theorem,’ Radon-Nikodym theorem, dominated convergence theorem, Fatou’s lemma, L^p-space, Holder’s inequality, conditional mathematical expectation, Jensen’s inequality, conditional independence, product probability space and Fubini’s theorem. In this chapter we give some supplements, necessary for reading this book.

Throughout this book we use the following common notations.

• N = the set of all non-negative integers.
• $\overline{N}=N\cup \{+\infty \}$.
• R = ] − ∞, + ∞[¹⁾ (real line).
• R ₊ = [0,∞[.
• $\overline{R}=[-\infty ,+\infty ]\left(\text{extended real line}\right)$.
• ${\overline{R}}_{+}=[0,+\infty ]$.
• Q (resp. Q ₊) = the set of all rationale in R (resp. R +).
• B(R ) ( resp.B(R ₊)) = the Borei σ-field in R (resp. R₊).

Let Ω be a set. A mapping from Ω to R (resp. $\overline{R}$) is called a (resp. extended) real-valued function on Ω. As restricted on Ω, a set always means a subset of Ω. The union and intersection of sets A and B are denoted by A ∪ B and A ∩ B (or simply AB) respectively. The complement of A is denoted by A^c. A\B = AB^c is the difference between A and B. A Δ B = (A\B) ∪ (B\A) is the symmetric difference of A and B. Empty set is denoted by ∅. The indicator of a set ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Dedication Page
• Chapter I Preliminaries
• Chapter II Classical Martingale Theory
• Chapter III Processes and Stopping Times
• Chapter IV Section Theorems and Their Applications
• Chapter V Projections of Processes
• Chapter VI Martingales with Integrable Variation and Square Integrable Martingales
• Chapter VII Local Martingales
• Chapter VIII Semimartingales and Quasimartingales
• Chapter IX Stochastic Integrals
• Chapter X Martingale Spaces H1 and BMO
• Chapter XI The Characteristics of Semimartingales
• Chapter XII Changes of Measures
• Chapter XIII Predictable Representation Property
• Chapter XIV Absolute Continuity and Contiguity of Measures
• Chapter XV Weak Convergence for Cadlag Processes
• Chapter XVI Weak Convergence for Semimartingales
• References
• Index