Understanding Markov Chains - Examples And Applications - Privault Nicolas

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Résumé :
pThis book provides an undergraduate introduction to discrete andï¿?continuous-time Markov chains and their applications. A large focus is placed on the first step analysisï¿?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./p

Biographie:
The author 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.

Sommaire:

Introduction

1 Probability Background 1.1 Probability Spaces and Events 1.2 Probability Measures 1.3 Conditional Probabilities and Independence 1.4 Random Variables 1.5 Probability Distributions 1.6 Expectation of a Random Variable 1.7 Conditional Expectation 1.8 Moment and Probability Generating Functions Exercises

2 Gambling Problems 2.1 Constrained Random Walk 2.2 Ruin Probabilities 2.3 Mean Game Duration Exercises

3 Random Walk 3.1 Unrestricted Random Walk 3.2 Mean and Variance 3.3 Distribution 3.4 First Return to Zero Exercises

4 Discrete-Time Markov Chains 4.1 Markov Property 4.2 Transition matrix 4.3 Examples of Markov Chains 4.4 Higher Order Transition Probabilities 4.5 The Two-State Discrete-Time Markov Chain Exercises

5 First Step Analysis 5.1 Hitting Probabilities 5.2 Mean Hitting and Absorption Times 5.3 First Return Times 5.4 Number of Returns Exercises

6 Classication of States 6.1 Communicating States 6.2 Recurrent States 6.3 Transient States 6.4 Positive and Null Recurrence 6.5 Periodicity and Aperiodicity Exercises

7 Long-Run Behavior of Markov Chains 7.1 Limiting Distributions 7.2 Stationary Distributions 7.3 Markov Chain Monte Carlo Exercises

8 Branching Processes 8.1 Defnition and Examples 8.2 Probability Generating Functions 8.3 Extinction Probabilities Exercises

9 Continuous-Time Markov Chains 9.1 The Poisson Process 9.2 Continuous-Time Chains 9.3 Transition Semigroup9.4 Infinitesimal Generator 9.5 The Two-State Continuous-Time Markov Chain 9.6 Limiting and Stationary Distributions 9.7 The Discrete-Time Embedded Chain 9.8 Mean Absorption Time and Probabilities Exercises

10 Discrete-Time Martingales 10.1 Filtrations and Conditional Expectations 10.2 Martingales - Definition and Properties 10.3 Ruin Probabilities 10.4 Mean Game Duration Exercises

11 Spatial Poisson Processes 11.1 Spatial Poisson (1781-1840) Processes 11.2 Poisson Stochastic Integrals 11.3 Transformations of Poisson Measures 11.4 Moments of Poisson Stochastic Integrals 11.5 Deviation Inequalities Exercises

12 Reliability Theory 12.1 Survival Probabilities 12.2 Poisson Process with Time-Dependent Intensity 12.3 Mean Time to Failure Exercises

Some Useful Identities Solutions to the Exercises References Index