© 2019

Nonlinear Expectations and Stochastic Calculus under Uncertainty

with Robust CLT and G-Brownian Motion


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

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Basic Theory of Nonlinear Expectations

  3. Stochastic Analysis Under G-Expectations

    1. Front Matter
      Pages 47-47
    2. Shige Peng
      Pages 101-112
  4. Stochastic Calculus for General Situations

    1. Front Matter
      Pages 145-145
    2. Shige Peng
      Pages 147-156
  5. Back Matter
    Pages 171-212

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.


probability theory stochastic analysis uncertainty of probabilities nonlinear expectations independence and identical distribution under uncertainty law of large numbers central limit theorem maximal distribution G-normal distribution G-Brownian motion G-martingale G-martingale representation theorem stochastic integral of G-Brownian motion stochastic differential equations driven by G-Brownian motion quadratic variation process of G-Brownian motion nonlinear Feynman-Kac formula mathematical statistics

Authors and affiliations

  1. 1.Institute of MathematicsShandong UniversityJinanChina

About the authors

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
  • Series Abbreviated Title Probability and Stochastic (formerly: PIA & SMAP)
  • DOI
  • Copyright Information Springer-Verlag GmbH Germany, part of Springer Nature 2019
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-662-59902-0
  • Softcover ISBN 978-3-662-59905-1
  • eBook ISBN 978-3-662-59903-7
  • Series ISSN 2199-3130
  • Series E-ISSN 2199-3149
  • Edition Number 1
  • Number of Pages XIII, 212
  • Number of Illustrations 10 b/w illustrations, 0 illustrations in colour
  • Topics Probability Theory and Stochastic Processes
    Quantitative Finance
  • Buy this book on publisher's site
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“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)