Book Description
A comprehensive and accessible presentation of probability and stochastic processes with emphasis on key theoretical concepts and realworld 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, semiMarkov 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
Chapterbychapter 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 upperundergraduate 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
 Cover
 Title Page
 Preface
 Acknowledgments
 Introduction

Part I: Probability
 Chapter 1: Elements of Probability Measure

Chapter 2: Random Variables
 Reduction to . Random variables
 2.1 Discrete and Continuous Random Variables
 2.2 Examples of Commonly Encountered Random Variables
 2.3 Existence of Random Variables with Prescribed Distribution. Skorohod Representation of a Random Variable
 2.4 Independence
 2.5 Functions of Random Variables. Calculating Distributions
 Problems
 Chapter 3: Applied Chapter: Generating Random Variables

Chapter 4: Integration Theory
 4.1 Integral of Measurable Functions
 4.2 Expectations
 4.3 Moments of a Random Variable. Variance and the Correlation Coefficient
 4.4 Functions of Random Variables. The Transport Formula
 4.5 Applications. Exercises in Probability Reasoning
 4.6 A Basic Central Limit Theorem: The DeMoivre–LaplaceTheorem:
 Problems
 Chapter 5: Product Spaces. Conditional Distribution and Conditional Expectation
 Chapter 6: Tools to study Probability. Generating Function, Moment Generating Function, Characteristic Function

Chapter 7: Limit Theorems
 Introduction
 7.1 Types of Convergence
 7.2 Relationships between Types of Convergence
 7.3 Continuous Mapping Theorem. Joint Convergence. Slutsky's Theorem
 7.4 The Two Big Limit Theorems: LLN and CLT
 7.5 Extensions of Central Limit Theorem. Limit Theorems for Other Types of Statistics
 7.6 Exchanging the Order of Limits and Expectations
 Problems
 Chapter 8: Statistical Inference

Part II: Stochastic Processes
 Chapter 9: Introduction to Stochastic Processes
 Chapter 10: The Poisson Process

Chapter 11: Renewal Processes
 11.1 Limit Theorems for the Renewal Process
 11.2 Discrete Renewal Theory. Blackwell Theorem
 11.3 The Key Renewal Theorem
 11.4 Applications of the Renewal Theorems
 11.5 Special cases of Renewal Processes. Alternating Renewal process. Renewal Reward process.
 11.6 A generalized approach. The Renewal Equation. Convolutions.
 11.7 AgeDependent Branching processes
 Problems
 Chapter 12: Markov Chains

Chapter 13: SemiMarkov Processes and Continuous time Markov Processes
 13.1 Characterization Theorems for the General semi Markov Process
 13.2 ContinuousTime Markov Processes
 13.3 The Kolmogorov Differential Equations
 13.4 Calculating Transition Probabilities for a Markov Process. General Approach
 13.5 Limiting Probabilities for the ContinuousTime Markov Chain
 13.6 Reversible Markov Process
 Problems
 Chapter 14: Martingales
 Chapter 15: Brownian Motion

Chapter 16: Stochastic Differential Equations with respect to Brownian Motion
 16.1 The Construction of the Stochastic Integral
 16.2 Properties of the Stochastic Integral
 16.3 Itô lemma
 16.4 Stochastic Differential Equations (SDEs)
 16.5 Examples of SDEs
 16.6 Linear Systems of SDEs
 16.7 A Simple Relationship between SDEs and Partial Differential Equations (PDEs)
 16.8 Monte Carlo Simulations of SDEs
 Problems

Appendix A: Appendix: Linear Algebra and Solving Difference Equations and Systems of Differential Equations
 A.1 Solving difference equations with constant coefficients
 A.2 Generalized matrix inverse and pseudodeterminant
 A.3 Connection between systems of differential equations and matrices
 A.4 Linear Algebra results
 A.5 Finding fundamental solution of the homogeneous system
 A.6 The nonhomogeneous system
 A.7 Solving systems when P is nonconstant
 Bibliography
 End User License Agreement
Product Information
 Title: Probability and Stochastic Processes
 Author(s):
 Release date: October 2014
 Publisher(s): Wiley
 ISBN: 9780470624555