Emphasizing fundamental mathematical ideas rather than proofs, Introduction to Stochastic Processes, Second Edition provides quick access to important foundations of probability theory applicable to problems in many fields. Assuming that you have a reasonable level of computer literacy, the ability to write simple programs, and the access to software for linear algebra computations, the author approaches the problems and theorems with a focus on stochastic processes evolving with time, rather than a particular emphasis on measure theory.
For those lacking in exposure to linear differential and difference equations, the author begins with a brief introduction to these concepts. He proceeds to discuss Markov chains, optimal stopping, martingales, and Brownian motion. The book concludes with a chapter on stochastic integration. The author supplies many basic, general examples and provides exercises at the end of each chapter.
New to the Second Edition:
Applicable to the fields of mathematics, statistics, and engineering as well as computer science, economics, business, biological science, psychology, and engineering, this concise introduction is an excellent resource both for students and professionals.
Preface to First Edition
PRELIMINARIES
Introduction
Linear Differential Equations
Linear Difference Equations
Exercises
FINITE MARKOV CHAINS
Definitions and Examples
Large-Time Behavior and Invariant Probability
Classification of States
Return Times
Transient States
Examples
Exercises
COUNTABLE MARKOV CHAINS
Introduction
Recurrence and Transience
Positive Recurrence and Null Recurrence
Branching Process
Exercises
CONTINUOUS-TIME MARKOV CHAINS
Poisson Process
Finite State Space
Birth-and-Death Processes
General Case
Exercises
OPTIMAL STOPPING
Optimal Stopping of Markov Chains
Optimal Stopping with Cost
Optimal Stopping with Discounting
Exercises
MARTINGALES
Conditional Expectation
Definition and Examples
Optional Sampling Theorem
Uniform Integrability
Martingale Convergence Theorem
Maximal Inequalities
Exercises
RENEWAL PROCESSES
Introduction
Renewal Equation
Discrete Renewal Processes
M/G/1 and G/M/1 Queues
Exercises
REVERSIBLE MARKOV CHAINS
Reversible Processes
Convergence to Equilibrium
Markov Chain Algorithms
A Criterion for Recurrence
Exercises
BROWNIAN MOTION
Introduction
Markov Property
Zero Set of Brownian Motion
Brownian Motion in Several Dimensions
Recurrence and Transience
Fractal Nature of Brownian Motion
Scaling Rules
Brownian Motion with Drift
Exercises
STOCHASTIC INTEGRATION
Integration with Respect to Random Walk
Integration with Respect to Brownian Motion
Itô's Formula
Extensions if Itô's Formula
Continuous Martingales
Girsanov Transformation
Feynman-Kac Formula
Black-Scholes Formula
Simulation
Exercises
Suggestions for Further Reading
Index
Biography
Greogory F. Lawler
