Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- 1 Sums of Independent Random Variables
- 2 The Central Limit Theorem
- 3 Infinitely Divisible Laws
- 4 Lévy Processes
- 5 Conditioning and Martingales
- 6 Some Extensions and Applications of Martingale Theory
- 7 Continuous Parameter Martingales
- 8 Gaussian Measures on a Banach Space
- 9 Convergence of Measures on a Polish Space
- 10 Wiener Measure and Partial Differential Equations
- 11 Some Classical Potential Theory
- References
- Index
4 - Lévy Processes
Published online by Cambridge University Press: 07 November 2024
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- 1 Sums of Independent Random Variables
- 2 The Central Limit Theorem
- 3 Infinitely Divisible Laws
- 4 Lévy Processes
- 5 Conditioning and Martingales
- 6 Some Extensions and Applications of Martingale Theory
- 7 Continuous Parameter Martingales
- 8 Gaussian Measures on a Banach Space
- 9 Convergence of Measures on a Polish Space
- 10 Wiener Measure and Partial Differential Equations
- 11 Some Classical Potential Theory
- References
- Index
Summary
In Chapter 4 I construct the Lévy processes (a.k.a. independent increment processes) corresponding to infinitely divisible laws. Section 4.1 provides the requisite information about the pathspace D(ℝN) of right-continuous paths with left limits, and §4.2 gives the construction of Lévy processes with discontinuous paths, the ones corresponding to infinitely divisible laws having no Gaussian part. Finally, in §4.3 I construct Brownian motion, the Lévy process with continuous paths, following the prescription given by Lévy. This section also contains a derivation of Kolmogorov’s continuity criterion for general Banach space-valued stochastic processes.
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- Information
- Probability Theory, An Analytic View , pp. 131 - 169Publisher: Cambridge University PressPrint publication year: 2024