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Nonlinear Expectations and Stochastic Calculus under Uncertainty

with Robust CLT and G-Brownian Motion

  • Book
  • © 2019

Overview

Authors:
  1. Shige Peng

    1. Institute of Mathematics, Shandong University, Jinan, China

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  • Provides new notions and results of the theory of nonlinear expectations and related stochastic analysis
  • Summarizes the latest studies on G-Martingale representation theorem and Itô’s integrals
  • Includes exercises that help reader master and learn in each chapter

Part of the book series: Probability Theory and Stochastic Modelling (PTSM, volume 95)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Basic Theory of Nonlinear Expectations

    1. Front Matter

      Pages 1-1
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    2. Sublinear Expectations and Risk Measures

      • Shige Peng
      Pages 3-21

    3. Law of Large Numbers and Central Limit Theorem Under Probability Uncertainty

      • Shige Peng
      Pages 23-45

  3. Stochastic Analysis Under G-Expectations

    1. Front Matter

      Pages 47-47
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    2. G-Brownian Motion and Itô’s Calculus

      • Shige Peng
      Pages 49-89

    3. G-Martingales and Jensen’s Inequality

      • Shige Peng
      Pages 91-100

    4. Stochastic Differential Equations

      • Shige Peng
      Pages 101-112

    5. Capacity and Quasi-surely Analysis for G-Brownian Paths

      • Shige Peng
      Pages 113-143

  4. Stochastic Calculus for General Situations

    1. Front Matter

      Pages 145-145
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    2. G-Martingale Representation Theorem

      • Shige Peng
      Pages 147-156

    3. Some Further Results of Itô’s Calculus

      • Shige Peng
      Pages 157-170

  5. Back Matter

    Pages 171-212
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Keywords

About this book

This book is focused on the recent developments on problems of probability model uncertainty by using the notion of nonlinear expectations and, in particular, sublinear expectations. It provides a gentle coverage of the theory of nonlinear expectations and related stochastic analysis. Many notions and results, for example, G-normal distribution, G-Brownian motion, G-Martingale representation theorem, and related stochastic calculus are first introduced or obtained by the author.

This book is based on Shige Peng’s lecture notes for a series of lectures given at summer schools and universities worldwide. It starts with basic definitions of nonlinear expectations and their relation to coherent measures of risk, law of large numbers and central limit theorems under nonlinear expectations, and develops into stochastic integral and stochastic calculus under G-expectations. It ends with recent research topic on G-Martingale representation theorem and G-stochastic integral for locally integrable processes.

With exercises to practice at the end of each chapter, this book can be used as a graduate textbook for students in probability theory and mathematical finance. Each chapter also concludes with a section Notes and Comments, which gives history and further references on the material covered in that chapter.

Researchers and graduate students interested in probability theory and mathematical finance will find this book very useful.

Reviews

“The book is very interesting and useful for the specialists in stochastic calculus and its financial and other applications. It is written in a very clear language and therefore can be used for graduate students and practitioners. It presents very recent and modern subjects and so it will find a wide audience.” (Yuliya S. Mishura, zbMATH 1427.60004, 2020)

Authors and Affiliations

  • Institute of Mathematics, Shandong University, Jinan, China

    Shige Peng

About the author

Shige Peng received his PhD in 1985 at Université Paris-Dauphine, in the direction of mathematics and informatics, and 1986 at University of Provence, in the direction of applied mathematics. He now is a full professor in Shandong University. His main research interests are stochastic optimal controls, backward SDEs and the corresponding PDEs, stochastic HJB equations. He has received the Natural Science Prize of China (1995), Su Buqing Prize of Applied Mathematics (2006), TAN Kah Kee Science Award (2008), Loo-Keng Hua Mathematics Award (2011), and the Qiu Shi Award for Outstanding Scientists (2016).

Bibliographic Information

  • Book Title: Nonlinear Expectations and Stochastic Calculus under Uncertainty

  • Book Subtitle: with Robust CLT and G-Brownian Motion

  • Authors: Shige Peng

  • Series Title: Probability Theory and Stochastic Modelling

  • DOI: https://doi.org/10.1007/978-3-662-59903-7

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag GmbH Germany, part of Springer Nature 2019

  • Hardcover ISBN: 978-3-662-59902-0Published: 19 September 2019

  • Softcover ISBN: 978-3-662-59905-1Published: 19 September 2020

  • eBook ISBN: 978-3-662-59903-7Published: 09 September 2019

  • Series ISSN: 2199-3130

  • Series E-ISSN: 2199-3149

  • Edition Number: 1

  • Number of Pages: XIII, 212

  • Number of Illustrations: 10 b/w illustrations

  • Topics: Probability Theory and Stochastic Processes, Quantitative Finance

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