Nonlinear Expectations and Stochastic Calculus under Uncertainty (eBook)

with Robust CLT and G-Brownian Motion

(Autor)

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2019 | 1st ed. 2019
XIII, 212 Seiten
Springer Berlin Heidelberg (Verlag)
978-3-662-59903-7 (ISBN)

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Nonlinear Expectations and Stochastic Calculus under Uncertainty - Shige Peng
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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.



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).

Preface 6
Introduction 8
Contents 12
Part I Basic Theory of Nonlinear Expectations 15
1 Sublinear Expectations and Risk Measures 16
1.1 Sublinear Expectations and Sublinear Expectation Spaces 16
1.2 Representation of a Sublinear Expectation 19
1.3 Distributions, Independence and Product Spaces 20
1.4 Completion of Sublinear Expectation Spaces 26
1.5 Examples of i.i.d Sequences Under Uncertainty of Probabilities 28
1.6 Relation with Coherent Measures of Risk 30
1.7 Exercises 32
2 Law of Large Numbers and Central Limit Theorem Under Probability Uncertainty 35
2.1 Some Basic Results of Parabolic Partial Differential Equations 35
2.2 Maximal Distribution and G-Normal Distribution 38
2.3 Existence of G-Distributed Random Variables 44
2.4 Law of Large Numbers and Central Limit Theorem 46
2.5 Exercises 54
Part II Stochastic Analysis Under G-Expectations 58
3 G-Brownian Motion and Itô's Calculus 59
3.1 Brownian Motion on a Sublinear Expectation Space 59
3.2 Existence of G-Brownian Motion 63
3.3 Itô's Integral with Respect to G-Brownian Motion 67
3.4 Quadratic Variation Process of G-Brownian Motion 70
3.5 Distribution of the Quadratic Variation Process langleB rangle 77
3.6 Itô's Formula 81
3.7 Brownian Motion Without Symmetric Condition 88
3.8 G-Brownian Motion Under (Not Necessarily Sublinear) Nonlinear Expectation 91
3.9 Construction of Brownian Motions on a Nonlinear Expectation Space 94
3.10 Exercises 97
4 G-Martingales and Jensen's Inequality 100
4.1 The Notion of G-Martingales 100
4.2 Heuristic Explanation of G-Martingale Representation 102
4.3 G-Convexity and Jensen's Inequality for G-Expectations 104
4.4 Exercises 108
5 Stochastic Differential Equations 110
5.1 Stochastic Differential Equations 110
5.2 Backward Stochastic Differential Equations (BSDE) 113
5.3 Nonlinear Feynman-Kac Formula 115
5.4 Exercises 119
6 Capacity and Quasi-surely Analysis for G-Brownian Paths 122
6.1 Integration Theory Associated to Upper Probabilities 122
6.1.1 Capacity Associated with P 123
6.1.2 Functional Spaces 126
6.1.3 Properties of Elements of mathbbLpc 130
6.1.4 Kolmogorov's Criterion 132
6.2 G-Expectation as an Upper Expectation 134
6.2.1 Construction of G-Brownian Motion Through Its Finite Dimensional Distributions 134
6.2.2 G-Expectation: A More Explicit Construction 136
6.3 The Capacity of G-Brownian Motion 143
6.4 Quasi-continuous Processes 147
6.5 Exercises 150
Part III Stochastic Calculus for General Situations 153
7 G-Martingale Representation Theorem 154
7.1 G-Martingale Representation Theorem 154
8 Some Further Results of Itô's Calculus 164
8.1 A Generalized Itô's Integral 164
8.2 Itô's Integral for Locally Integrable Processes 170
8.3 Itô's Formula for General C2 Functions 174
Appendix A Preliminaries in Functional Analysis 178
A.1 Completion of Normed Linear Spaces 178
A.2 The Hahn-Banach Extension Theorem 179
A.3 Dini's Theorem and Tietze's Extension Theorem 179
Appendix B Preliminaries in Probability Theory 180
B.1 Kolmogorov's Extension Theorem 180
B.2 Kolmogorov's Criterion 181
B.3 Daniell-Stone Theorem 183
B.4 Some Important Inequalities 183
Appendix C Solutions of Parabolic Partial Differential Equation 185
C.1 The Definition of Viscosity Solutions 185
C.2 Comparison Theorem 187
C.3 Perron's Method and Existence 196
C.4 Krylov's Regularity Estimate for Parabolic PDEs 201
Appendix References 204
Appendix Index of Symbols 211
Appendix Author Index 213
Author Index 213
Appendix Subject Index 215
Index 215

Erscheint lt. Verlag 9.9.2019
Reihe/Serie Probability Theory and Stochastic Modelling
Probability Theory and Stochastic Modelling
Zusatzinfo XIII, 212 p. 10 illus.
Sprache englisch
Themenwelt Mathematik / Informatik Mathematik Statistik
Mathematik / Informatik Mathematik Wahrscheinlichkeit / Kombinatorik
Schlagworte central limit theorem • G-Brownian motion • G-martingale • G-martingale representation theorem • G-normal distribution • independence and identical distribution under uncertainty • law of large numbers • Mathematical Statistics • maximal distribution • nonlinear expectations • nonlinear Feynman-Kac formula • Probability Theory • quadratic variation process of G-Brownian motion • Quantitative Finance • stochastic analysis • stochastic differential equations driven by G-Brownian motion • stochastic integral of G-Brownian motion • uncertainty of probabilities
ISBN-10 3-662-59903-1 / 3662599031
ISBN-13 978-3-662-59903-7 / 9783662599037
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