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© 2017

Stochastic Modeling

Benefits

  • Contains 175 exercises including research-oriented problems about special stochastic processes not covered in traditional textbooks

  • Includes detailed simulation programs of the main models

  • Covers topics not typically included in traditional textbooks, allowing for readers to learn quickly on many topics, including research-oriented topics

  • Includes a timeline with the main contributors since the origin of probability theory until today

Textbook

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Probability theory

    1. Front Matter
      Pages 1-1
    2. Nicolas Lanchier
      Pages 3-24
    3. Nicolas Lanchier
      Pages 25-40
    4. Nicolas Lanchier
      Pages 41-56
  3. Stochastic processes

    1. Front Matter
      Pages 57-57
    2. Nicolas Lanchier
      Pages 59-63
    3. Nicolas Lanchier
      Pages 65-91
    4. Nicolas Lanchier
      Pages 93-99
    5. Nicolas Lanchier
      Pages 101-128
    6. Nicolas Lanchier
      Pages 129-139
    7. Nicolas Lanchier
      Pages 141-160
    8. Nicolas Lanchier
      Pages 161-189
  4. Special models

    1. Front Matter
      Pages 191-191
    2. Nicolas Lanchier
      Pages 193-201
    3. Nicolas Lanchier
      Pages 203-218
    4. Nicolas Lanchier
      Pages 219-234
    5. Nicolas Lanchier
      Pages 235-244
    6. Nicolas Lanchier
      Pages 245-258
    7. Nicolas Lanchier
      Pages 259-268

About this book

Introduction

Three coherent parts form the material covered in this text, portions of which have not been widely covered in traditional textbooks. In this coverage the reader is quickly introduced to several different topics enriched with 175 exercises which focus on real-world problems. Exercises range from the classics of probability theory to more exotic research-oriented problems based on numerical simulations. Intended for graduate students in mathematics and applied sciences, the text provides the tools and training needed to write and use programs for research purposes.

The first part of the text begins with a brief review of measure theory and revisits the main concepts of probability theory, from random variables to the standard limit theorems. The second part covers traditional material on stochastic processes, including martingales, discrete-time Markov chains, Poisson processes, and continuous-time Markov chains. The theory developed is illustrated by a variety of examples surrounding applications such as the gambler’s ruin chain, branching processes, symmetric random walks, and queueing systems. The third, more research-oriented part of the text, discusses special stochastic processes of interest in physics, biology, and sociology. Additional emphasis is placed on minimal models that have been used historically to develop new mathematical techniques in the field of stochastic processes: the logistic growth process, the Wright–Fisher model, Kingman’s coalescent, percolation models, the contact process, and the voter model. Further treatment of the material explains how these special processes are connected to each other from a modeling perspective as well as their simulation capabilities in C and Matlab™.

Keywords

Martingales Markov Chains Poisson Processes Symmetric Random Walks Branching Processes Wright-Fisher Model Percolation Models Contact Process Vorter Model Numerical Simulations

Authors and affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA

About the authors

Nicolas Lanchier is Associate Professor at Arizona State University, School of Mathematical and Statistical Sciences.  His research interests include mathematical models introduced in the life and social sciences literature that describe inherently spatial phenomena of interacting populations consist of systems of ordinary differential equations.  

Bibliographic information

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Reviews

“Stochastic Modeling by Nicolas Lanchier is an introduction to stochastic processes accessible to advanced students and interdisciplinary scientists with a background in graduate-level real analysis. The work offers a rigorous approach to stochastic models used in social, biological and physical sciences ... . Stochastic modeling provides a link between applied research in stochastic models and the literature covering their mathematical foundations.”  (Ben Dyhr, Mathematical Reviews, May, 2018)

“There is a wide spectrum of topics discussed in this book. … It is also interesting to find several classical examples with all details. … The text is so carefully written and checked, that I was unable to find a single typo. The book can be strongly recommended to those students and teachers who want to be in line with modern probability theory and its diverse applications.” (Jordan M. Stoyanov, zbMATH 1360.60002, 2017)