Diffusion Processes, Jump Processes, and Stochastic Differential Equations

Wojbor A. Woyczyński

• Publisher: Chapman and Hall/CRC
• Year: 2022
• eBook ISBN: 9781000475371
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Diffusion Processes, Jump Processes, and Stochastic Differential Equations provides a compact exposition of the results explaining interrelations between diffusion stochastic processes, stochastic differential equations and the fractional infinitesimal operators. The draft of this book has been extensively classroom tested by the author at Case Western Reserve University in a course that enrolled seniors and graduate students majoring in mathematics, statistics, engineering, physics, chemistry, economics and mathematical finance. The last topic proved to be particularly popular among students looking for careers on Wall Street and in research organizations devoted to financial problems.

Features

• Quickly and concisely builds from basic probability theory to advanced topics
• Suitable as a primary text for an advanced course in diffusion processes and stochastic differential equations
• Useful as supplementary reading across a range of topics.

Chapter 1 Random Variables, Vectors, Processes, and Fields

DOI: 10.1201/9781003216759-1

1.1 Random Variables, Vectors, and Their Distributions—A Glossary

1.1.1 Basic Concepts

We will denote by (Ω, ℱ, P) the probability space (triple) consisting of the sample space Ω, the σ-algebra ℱ of subsets of Ω, and a probability measure P on ℱ. Elements ω ∈ Ω will be called sample points, and elements of ℱ, random events. The probability measure P is assumed to be non-negative, countably additive, i.e., for a pairwise disjoint sequence A_n, n − 1, 2, . . ., of random events,

$$\mathbf{\text{P}}\left({\displaystyle \underset{n=1}{\overset{\infty }{\cup }}{A}_n}\right)={\displaystyle \sum _{n=1}^{\infty }\mathbf{\text{P}}}\left(A_n\right),$$

and normalized, i.e., P(Ω) = 1.

A (real-valued) random variable is a mapping X : Ω → R which is measurable, that is, a mapping such that, for any x ∈ R, the set of sample points

$$\{\omega :X(\omega )\le x\}=X^{-1}((-\infty ,x])$$

is a random event, and its probability is well defined. In other words, for X to be a random variable, ∀_x ∈ R, the set {ω : X(ω) ≤ x} must be a member of the σ-field of random events ℱ. The set of all real-valued random variables will be denoted by L₀(Ω, ℱ, P; R).

To each random variable X, we attach its distribution, μ_X, that is a measure on the Borel subsets B ∈ R, defined by the equality

$${\mu }_X(B):=\mathbf{\text{P}}(X^{-1}(B))=\mathbf{\text{P}}(\omega :X(\omega )\in B)=\mathbf{\text{P}}(X\in B)\mathrm{.}$$

Recall that the Bo...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Author
• Chapter 1 ■ Random Variables, Vectors, Processes, and Fields
• Chapter 2 ■ From Random Walk to Brownian Motion
• Chapter 3 ■ Poisson Processes and Their Mixtures
• Chapter 4 ■ Lévy Processes and the Lévy-Khinchine Formula: Basic Facts
• Chapter 5 ■ General Processes with Independent Increments
• Chapter 6 ■ Stochastic Integrals for Brownian Motion and General Lévy Processes
• Chapter 7 ■ Itô Stochastic Differential Equations
• Chapter 8 ■ Asymmetric Exclusion Processes and Their Scaling Limits
• Chapter 9 ■ Nonlinear Diffusion Equations
• Appendix A ■ The Remarkable Bernoulli Family
• Bibliography
• Index