Introduction to Stochastic Processes, Second Edition

by Lawler; Gregory F.

• ISBN13: 9781584886518
• ISBN10: 158488651X
• Edition: 2nd
• Format: Hardcover
• Copyright: 2006-05-16
• Publisher: Chapman & Hall/

Summary

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:· Expanded chapter on stochastic integration that introduces modern mathematical finance· Introduction of Girsanov transformation and the Feynman-Kac formula· Expanded discussion of Itô's formula and the Black-Scholes formula for pricing options· New topics such as Doob's maximal inequality and a discussion on self similarity in the chapter on Brownian motionApplicable 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.

Table of Contents

Preface to Second Edition	ix
Preface to First Edition	xi
Preliminaries	1	(8)
Introduction	1	(1)
Linear Differential Equations	1	(2)
Linear Difference Equations	3	(3)
Exercises	6	(3)
Finite Markov Chains	9	(34)
Definitions and Examples	9	(5)
Large-Time Behavior and Invariant Probability	14	(3)
Classification of States	17	(7)
Reducibility	19	(2)
Periodicity	21	(1)
Irreducible, aperiodic chains	22	(1)
Reducible or periodic chains	22	(2)
Return Times	24	(2)
Transient States	26	(5)
Examples	31	(4)
Exercises	35	(8)
Countable Markov Chains	43	(22)
Introduction	43	(2)
Recurrence and Transience	45	(5)
Positive Recurrence and Null Recurrence	50	(3)
Branching Process	53	(4)
Exercises	57	(8)
Continuous-Time Markov Chains	65	(22)
Poisson Process	65	(3)
Finite State Space	68	(6)
Birth-and-Death Processes	74	(7)
General Case	81	(1)
Exercises	82	(5)
Optimal Stopping	87	(14)
Optimal Stopping of Markov Chains	87	(6)
Optimal Stopping with Cost	93	(3)
Optimal Stopping with Discounting	96	(2)
Exercises	98	(3)
Martingales	101	(30)
Conditional Expectation	101	(5)
Definition and Examples	106	(4)
Optional Sampling Theorem	110	(4)
Uniform Integrability	114	(2)
Martingale Convergence Theorem	116	(6)
Maximal Inequalities	122	(3)
Exercises	125	(6)
Renewal Processes	131	(24)
Introduction	131	(5)
Renewal Equation	136	(8)
Discrete Renewal Processes	144	(4)
M/G/1 and G/M/1 Queues	148	(3)
Exercises	151	(4)
Reversible Markov Chains	155	(18)
Reversible Processes	155	(2)
Convergence to Equilibrium	157	(5)
Markov Chain Algorithms	162	(4)
A Criterion for Recurrence	166	(4)
Exercises	170	(3)
Brownian Motion	173	(26)
Introduction	173	(3)
Markov Property	176	(5)
Zero Set of Brownian Motion	181	(3)
Brownian Motion in Several Dimensions	184	(5)
Recurrence and Transience	189	(2)
Fractal Nature of Brownian Motion	191	(1)
Scaling Rules	192	(1)
Brownian Motion with Drift	193	(2)
Exercises	195	(4)
Stochastic Integration	199	(32)
Integration with Respect to Random Walk	199	(1)
Integration with Respect to Brownian Motion	200	(5)
Ito's Formula	205	(4)
Extensions of Ito's Formula	209	(7)
Continuous Martingales	216	(2)
Girsanov Transformation	218	(3)
Feynman-Kac Formula	221	(2)
Black-Scholes Formula	223	(5)
Simulation	228	(1)
Exercises	228	(3)
Suggestions for Further Reading	231	(2)
Index	233