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Probability and Stochastic Processes

Book Description

A comprehensive and accessible presentation of probability and stochastic processes with emphasis on key theoretical concepts and real-world applications

With a sophisticated approach, Probability and Stochastic Processes successfully balances theory and applications in a pedagogical and accessible format. The book's primary focus is on key theoretical notions in probability to provide a foundation for understanding concepts and examples related to stochastic processes.

Organized into two main sections, the book begins by developing probability theory with topical coverage on probability measure; random variables; integration theory; product spaces, conditional distribution, and conditional expectations; and limit theorems. The second part explores stochastic processes and related concepts including the Poisson process, renewal processes, Markov chains, semi-Markov processes, martingales, and Brownian motion. Featuring a logical combination of traditional and complex theories as well as practices, Probability and Stochastic Processes also includes:

  • Multiple examples from disciplines such as business, mathematical finance, and engineering

  • Chapter-by-chapter exercises and examples to allow readers to test their comprehension of the presented material

  • A rigorous treatment of all probability and stochastic processes concepts

  • An appropriate textbook for probability and stochastic processes courses at the upper-undergraduate and graduate level in mathematics, business, and electrical engineering, Probability and Stochastic Processes is also an ideal reference for researchers and practitioners in the fields of mathematics, engineering, and finance.

    Table of Contents

    1. Cover
    2. Title Page
      1. Copyright
      2. Dedication
    3. Preface
    4. Acknowledgments
    5. Introduction
    6. Part I: Probability
      1. Chapter 1: Elements of Probability Measure
        1. 1.1 Probability Spaces
        2. 1.2 Conditional Probability
        3. 1.3 Independence
        4. 1.4 Monotone Convergence Properties of Probability
        5. 1.5 Lebesgue Measure on the Unit Interval (0,1]
        6. Problems
      2. Chapter 2: Random Variables
        1. Reduction to <img xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0" src="images/c02-math-0020.gif" alt="c02-math-0020" style="vertical-align:middle;"></img>. Random variables. Random variables
        2. 2.1 Discrete and Continuous Random Variables
        3. 2.2 Examples of Commonly Encountered Random Variables
        4. 2.3 Existence of Random Variables with Prescribed Distribution. Skorohod Representation of a Random Variable
        5. 2.4 Independence
        6. 2.5 Functions of Random Variables. Calculating Distributions
        7. Problems
      3. Chapter 3: Applied Chapter: Generating Random Variables
        1. 3.1 Generating One-Dimensional Random Variables by Inverting the cdf
        2. 3.2 Generating One-Dimensional Normal Random Variables
        3. 3.3 Generating Random Variables. Rejection Sampling Method
        4. 3.4 Generating Random Variables. Importance Sampling
        5. Problems
      4. Chapter 4: Integration Theory
        1. 4.1 Integral of Measurable Functions
        2. 4.2 Expectations
        3. 4.3 Moments of a Random Variable. Variance and the Correlation Coefficient
        4. 4.4 Functions of Random Variables. The Transport Formula
        5. 4.5 Applications. Exercises in Probability Reasoning
        6. 4.6 A Basic Central Limit Theorem: <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0">The DeMoivre&#8211;LaplaceTheorem:</i>
        7. Problems
      5. Chapter 5: Product Spaces. Conditional Distribution and Conditional Expectation
        1. 5.1 Product Spaces
        2. 5.2 Conditional Distribution and Expectation. Calculation in Simple Cases
        3. 5.3 Conditional Expectation. General Definition
        4. 5.4 Random Vectors. Moments and Distributions
        5. Problems
      6. Chapter 6: Tools to study Probability. Generating Function, Moment Generating Function, Characteristic Function
        1. 6.1 Sums of Random Variables. Convolutions
        2. 6.2 Generating Functions and Applications
        3. 6.3 Moment Generating Function
        4. 6.4 Characteristic Function
        5. 6.5 Inversion and Continuity Theorems
        6. 6.6 Stable Distributions. Lévy Distribution
        7. Problems
      7. Chapter 7: Limit Theorems
        1. Introduction
        2. 7.1 Types of Convergence
        3. 7.2 Relationships between Types of Convergence
        4. 7.3 Continuous Mapping Theorem. Joint Convergence. Slutsky's Theorem
        5. 7.4 The Two Big Limit Theorems: LLN and CLT
        6. 7.5 Extensions of Central Limit Theorem. Limit Theorems for Other Types of Statistics
        7. 7.6 Exchanging the Order of Limits and Expectations
        8. Problems
      8. Chapter 8: Statistical Inference
        1. 8.1 The Classical Problems in Statistics
        2. 8.2 Parameter Estimation Problem
        3. 8.3 Maximum Likelihood Estimation Method
        4. 8.4 The Method of Moments
        5. 8.5 Testing, the Likelihood Ratio Test
        6. 8.6 Confidence Sets
        7. Problems
    7. Part II: Stochastic Processes
      1. Chapter 9: Introduction to Stochastic Processes
        1. 9.1 General Characteristics of Stochastic Processes
        2. 9.2 A Simple Process—The Bernoulli Process
        3. Problems
      2. Chapter 10: The Poisson Process
        1. Introduction
        2. 10.1 Definitions
        3. 10.2 Inter-Arrival and Waiting Time for a Poisson Process
        4. 10.3 General Poisson Processes
        5. 10.4 Simulation Techniques. Constructing the Poisson Process.
        6. Problems
      3. Chapter 11: Renewal Processes
        1. 11.1 Limit Theorems for the Renewal Process
        2. 11.2 Discrete Renewal Theory. Blackwell Theorem
        3. 11.3 The Key Renewal Theorem
        4. 11.4 Applications of the Renewal Theorems
        5. 11.5 Special cases of Renewal Processes. Alternating Renewal process. Renewal Reward process.
        6. 11.6 A generalized approach. The Renewal Equation. Convolutions.
        7. 11.7 Age-Dependent Branching processes
        8. Problems
      4. Chapter 12: Markov Chains
        1. 12.1 Basic Concepts for Markov Chains
        2. 12.2 Simple Random Walk on Integers in <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0">d</i> Dimensions Dimensions
        3. 12.3 Limit Theorems
        4. 12.4 Characterization of States for a Markov Chain. Stationary Distribution.
        5. 12.5 Other Issues: Graphs, First-Step Analysis
        6. 12.6 A general Treatment of the Markov Chains
        7. Problems
      5. Chapter 13: Semi-Markov Processes and Continuous time Markov Processes
        1. 13.1 Characterization Theorems for the General semi- Markov Process
        2. 13.2 Continuous-Time Markov Processes
        3. 13.3 The Kolmogorov Differential Equations
        4. 13.4 Calculating Transition Probabilities for a Markov Process. General Approach
        5. 13.5 Limiting Probabilities for the Continuous-Time Markov Chain
        6. 13.6 Reversible Markov Process
        7. Problems
      6. Chapter 14: Martingales
        1. 14.1 Definition and Examples
        2. 14.2 Martingales and Markov Chains
        3. 14.3 Previsible Process. The Martingale Transform
        4. 14.4 Stopping Time. Stopped Process
        5. 14.5 Classical Examples of Martingale Reasoning
        6. 14.6 Convergence Theorems. <img xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0" src="images/c14-math-0519.gif" alt="c14-math-0519" style="vertical-align:middle;"></img> Convergence. Bounded Martingales in Convergence. Bounded Martingales in <img xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0" src="images/c14-math-0520.gif" alt="c14-math-0520" style="vertical-align:middle;"></img>
        7. Problems
      7. Chapter 15: Brownian Motion
        1. 15.1 History
        2. 15.2 Definition
        3. 15.3 Properties of Brownian Motion
        4. 15.4 Simulating Brownian Motions
        5. Problems
      8. Chapter 16: Stochastic Differential Equations with respect to Brownian Motion
        1. 16.1 The Construction of the Stochastic Integral
        2. 16.2 Properties of the Stochastic Integral
        3. 16.3 Itô lemma
        4. 16.4 Stochastic Differential Equations (SDEs)
        5. 16.5 Examples of SDEs
        6. 16.6 Linear Systems of SDEs
        7. 16.7 A Simple Relationship between SDEs and Partial Differential Equations (PDEs)
        8. 16.8 Monte Carlo Simulations of SDEs
        9. Problems
    8. Appendix A: Appendix: Linear Algebra and Solving Difference Equations and Systems of Differential Equations
      1. A.1 Solving difference equations with constant coefficients
      2. A.2 Generalized matrix inverse and pseudo-determinant
      3. A.3 Connection between systems of differential equations and matrices
      4. A.4 Linear Algebra results
      5. A.5 Finding fundamental solution of the homogeneous system
      6. A.6 The nonhomogeneous system
      7. A.7 Solving systems when <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0">P</i> is non-constant is non-constant
    9. Bibliography
      1. Index
    10. End User License Agreement