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E-raamat: Understanding Markov Chains: Examples and Applications

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Understanding Markov Chains: Examples and ApplicationsSuurem pilt
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This book provides an undergraduate introduction to discrete and continuous-time Markov chains and their applications. A large focus is placed on the first step analysis technique and its applications to average hitting times and ruin probabilities. Classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes, are also covered. Two major examples (gambling processes and random walks) are treated in detail from the beginning, before the general theory itself is presented in the subsequent chapters. An introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times is also provided, and the book includes a chapter on spatial Poisson processes with some recent results on moment identities and deviation inequalities for Poisson stochastic integrals. The concepts presented are illustrated by examples and by 72 exercises and their complete solutions.

This book provides an undergraduate introduction to discrete and continuous-time Markov chains and their applications. It includes more than 70 exercises, along with complete solutions, that help illustrate and present all concepts.

Arvustused

This textbook provides an elementary introduction to the classical theory of discrete and continuous time Markov chains motivated by gambling problems and covers a variety of primers on different topics . this text may serve very well for a first undergraduate course on Markov chains for applied mathematicians, but also for students of financial engineering. It is completed by almost a hundred pages of solutions of exercises. (Michael Högele, zbMATH 1305.60003, 2015)

The book provides an introduction to discrete and continuous-time Markov chains and their applications. The explanation is detailed and clear. Often the reader is guided through the less trivial concepts by means of appropriate examples and additional comments, including diagrams and graphs. Also, a big plus is the presence of numerous well-chosen exercises at the end of each chapter, which are discussed in a separate `Solutions to the Exercises part at the end of the book. (Michele Zito, Mathematical Reviews, December, 2014)

1 Introduction
1(6)
2 Probability Background
7(30)
2.1 Probability Spaces and Events
7(4)
2.2 Probability Measures
11(1)
2.3 Conditional Probabilities and Independence
11(2)
2.4 Random Variables
13(2)
2.5 Probability Distributions
15(6)
2.6 Expectation of a Random Variable
21(6)
2.7 Conditional Expectation
27(2)
2.8 Moment and Probability Generating Functions
29(8)
3 Gambling Problems
37(24)
3.1 Constrained Random Walk
37(1)
3.2 Ruin Probabilities
38(11)
3.3 Mean Game Duration
49(12)
4 Random Walks
61(16)
4.1 Unrestricted Random Walk
61(1)
4.2 Mean and Variance
62(1)
4.3 Distribution
63(1)
4.4 First Return to Zero
64(13)
5 Discrete-Time Markov Chains
77(18)
5.1 Markov Property
77(2)
5.2 Transition Matrix
79(2)
5.3 Examples of Markov Chains
81(3)
5.4 Higher Order Transition Probabilities
84(3)
5.5 The Two-State Discrete-Time Markov Chain
87(8)
6 First Step Analysis
95(22)
6.1 Hitting Probabilities
95(3)
6.2 Mean Hitting and Absorption Times
98(5)
6.3 First Return Times
103(5)
6.4 Number of Returns
108(9)
7 Classification of States
117(12)
7.1 Communicating States
117(2)
7.2 Recurrent States
119(2)
7.3 Transient States
121(3)
7.4 Positive and Null Recurrence
124(1)
7.5 Periodicity and Aperiodicity
125(4)
8 Long-Run Behavior of Markov Chains
129(20)
8.1 Limiting Distributions
129(1)
8.2 Stationary Distributions
130(9)
8.3 Markov Chain Monte Carlo
139(10)
9 Branching Processes
149(18)
9.1 Definition and Examples
149(3)
9.2 Probability Generating Functions
152(2)
9.3 Extinction Probabilities
154(13)
10 Continuous-Time Markov Chains
167(44)
10.1 The Poisson Process
167(5)
10.2 Continuous-Time Chains
172(4)
10.3 Transition Semigroup
176(4)
10.4 Infinitesimal Generator
180(7)
10.5 The Two-State Continuous-Time Markov Chain
187(4)
10.6 Limiting and Stationary Distributions
191(5)
10.7 The Discrete-Time Embedded Chain
196(4)
10.8 Mean Absorption Time and Probabilities
200(11)
11 Discrete-Time Martingales
211(14)
11.1 Filtrations and Conditional Expectations
211(1)
11.2 Martingales---Definition and Properties
212(4)
11.3 Ruin Probabilities
216(4)
11.4 Mean Game Duration
220(5)
12 Spatial Poisson Processes
225(16)
12.1 Spatial Poisson (1781-1840) Processes
225(2)
12.2 Poisson Stochastic Integrals
227(2)
12.3 Transformations of Poisson Measures
229(2)
12.4 Moments of Poisson Stochastic Integrals
231(4)
12.5 Deviation Inequalities
235(6)
13 Reliability Theory
241(10)
13.1 Survival Probabilities
241(2)
13.2 Poisson Process with Time-Dependent Intensity
243(1)
13.3 Mean Time to Failure
244(7)
Some Useful Identities
247(4)
Solutions to the Exercises
251(98)
Chapter 2 Probability Background
251(9)
Chapter 3 Gambling Problems
260(6)
Chapter 4 Random Walks
266(5)
Chapter 5 Discrete-Time Markov Chains
271(1)
Chapter 6 First Step Analysis
272(18)
Chapter 7 Classification of States
290(2)
Chapter 8 Limiting and Stationary Distributions
292(20)
Chapter 9 Branching Processes
312(9)
Chapter 10 Continuous-Time Markov Chains
321(24)
Chapter 11 Discrete-Time Martingales
345(1)
Chapter 12 Spatial Poisson Processes
346(2)
Chapter 13 Reliability and Renewal Processes
348(1)
Bibliography 349(2)
Index 351
Nicolas Privault is an associate professor from the Nanyang Technological University (NTU) and is well-established in the field of stochastic processes and a highly respected probabilist. He has authored the book, Stochastic Analysis in Discrete and Continuous Settings: With Normal Martingales, Lecture Notes in Mathematics, Springer, 2009 and was a co-editor for the book, Stochastic Analysis with Financial Applications, Progress in Probability, Vol. 65, Springer Basel, 2011. Aside from these two Springer titles, he has authored several others. He is currently teaching the course M27004-Probability Theory and Stochastic Processes at NTU. The manuscript has been developed over the years from his courses on Stochastic Processes.
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