
Published online:
19 October 2017
Published in print:
21 July 2014
Online ISBN:
9781400850327
Print ISBN:
9780691161433
Contents
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1.1 Diffusions 1.1 Diffusions
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1.1.1 The Brownian Motion 1.1.1 The Brownian Motion
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1.1.2 Stochastic Integrals 1.1.2 Stochastic Integrals
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1.1.3 A Central Example: Diffusion Processes 1.1.3 A Central Example: Diffusion Processes
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1.2 Lévy Processes 1.2 Lévy Processes
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1.2.1 The Law of a Lévy Process 1.2.1 The Law of a Lévy Process
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1.2.2 Examples 1.2.2 Examples
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1.2.3 Poisson Random Measures 1.2.3 Poisson Random Measures
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1.2.4 Integrals with Respect to Poisson Random Measures 1.2.4 Integrals with Respect to Poisson Random Measures
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1.2.5 Path Properties and Lévy-Itô Decomposition 1.2.5 Path Properties and Lévy-Itô Decomposition
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1.3 Semimartingales 1.3 Semimartingales
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1.3.1 Definition and Stochastic Integrals 1.3.1 Definition and Stochastic Integrals
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1.3.2 Quadratic Variation 1.3.2 Quadratic Variation
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1.3.3 Itô’s Formula 1.3.3 Itô’s Formula
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1.3.4 Characteristics of a Semimartingale and the Lévy-Itô Decomposition 1.3.4 Characteristics of a Semimartingale and the Lévy-Itô Decomposition
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1.4 Itô Semimartingales 1.4 Itô Semimartingales
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1.4.1 The Definition 1.4.1 The Definition
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1.4.2 Extension of the Probability Space 1.4.2 Extension of the Probability Space
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1.4.3 The Grigelionis Form of an Itô Semimartingale 1.4.3 The Grigelionis Form of an Itô Semimartingale
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1.4.4 A Fundamental Example: Stochastic Differential Equations Driven by a Lévy Process 1.4.4 A Fundamental Example: Stochastic Differential Equations Driven by a Lévy Process
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1.5 Processes with Conditionally Independent Increments 1.5 Processes with Conditionally Independent Increments
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1.5.1 Processes with Independent Increments 1.5.1 Processes with Independent Increments
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1.5.2 A Class of Processes with Conditionally Independent Increments 1.5.2 A Class of Processes with Conditionally Independent Increments
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Cite
Aïıt-Sahalia, Yacine, and Jean Jacod, 'From Diffusions to Semimartingales', High-Frequency Financial Econometrics (Princeton, NJ , 2014; online edn, Princeton Scholarship Online, 19 Oct. 2017), https://doi.org/10.23943/princeton/9780691161433.003.0001, accessed 24 Apr. 2025.
Abstract
This chapter presents a quick review of the theory of semimartingales, which are processes for which statistical methods are considered in this book. Topics covered include diffusions, Lévy processes, Itô semimartingales, and processes with conditionally independent increments.
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