Peng Nonlinear Expectations and Stochastic Calculus under Uncertainty

with Robust CLT and G-Brownian Motion
1. Auflage 2019
ISBN: 978-3-662-59903-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 95, 216 Seiten

Reihe: Probability Theory and Stochastic Modelling

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).

Autoren/Hrsg.

Peng, Shige

Inhaltsverzeichnis

1;Preface;6
2;Introduction;8
3;Contents;12
4;Part I Basic Theory of Nonlinear Expectations;15
5;1 Sublinear Expectations and Risk Measures;16
5.1;1.1 Sublinear Expectations and Sublinear Expectation Spaces;16
5.2;1.2 Representation of a Sublinear Expectation;19
5.3;1.3 Distributions, Independence and Product Spaces;20
5.4;1.4 Completion of Sublinear Expectation Spaces;26
5.5;1.5 Examples of i.i.d Sequences Under Uncertainty of Probabilities;28
5.6;1.6 Relation with Coherent Measures of Risk;30
5.7;1.7 Exercises;32
6;2 Law of Large Numbers and Central Limit Theorem Under Probability Uncertainty;35
6.1;2.1 Some Basic Results of Parabolic Partial Differential Equations;35
6.2;2.2 Maximal Distribution and G-Normal Distribution;38
6.3;2.3 Existence of G-Distributed Random Variables;44
6.4;2.4 Law of Large Numbers and Central Limit Theorem;46
6.5;2.5 Exercises;54
7;Part II Stochastic Analysis Under G-Expectations;58
8;3 G-Brownian Motion and Itô's Calculus;59
8.1;3.1 Brownian Motion on a Sublinear Expectation Space;59
8.2;3.2 Existence of G-Brownian Motion;63
8.3;3.3 Itô's Integral with Respect to G-Brownian Motion;67
8.4;3.4 Quadratic Variation Process of G-Brownian Motion;70
8.5;3.5 Distribution of the Quadratic Variation Process langleB rangle;77
8.6;3.6 Itô's Formula;81
8.7;3.7 Brownian Motion Without Symmetric Condition;88
8.8;3.8 G-Brownian Motion Under (Not Necessarily Sublinear) Nonlinear Expectation;91
8.9;3.9 Construction of Brownian Motions on a Nonlinear Expectation Space;94
8.10;3.10 Exercises;97
9;4 G-Martingales and Jensen's Inequality;100
9.1;4.1 The Notion of G-Martingales;100
9.2;4.2 Heuristic Explanation of G-Martingale Representation;102
9.3;4.3 G-Convexity and Jensen's Inequality for G-Expectations;104
9.4;4.4 Exercises;108
10;5 Stochastic Differential Equations;110
10.1;5.1 Stochastic Differential Equations;110
10.2;5.2 Backward Stochastic Differential Equations (BSDE);113
10.3;5.3 Nonlinear Feynman-Kac Formula;115
10.4;5.4 Exercises;119
11;6 Capacity and Quasi-surely Analysis for G-Brownian Paths;122
11.1;6.1 Integration Theory Associated to Upper Probabilities;122
11.1.1;6.1.1 Capacity Associated with P;123
11.1.2;6.1.2 Functional Spaces;126
11.1.3;6.1.3 Properties of Elements of mathbbLpc;130
11.1.4;6.1.4 Kolmogorov's Criterion;132
11.2;6.2 G-Expectation as an Upper Expectation;134
11.2.1;6.2.1 Construction of G-Brownian Motion Through Its Finite Dimensional Distributions;134
11.2.2;6.2.2 G-Expectation: A More Explicit Construction;136
11.3;6.3 The Capacity of G-Brownian Motion;143
11.4;6.4 Quasi-continuous Processes;147
11.5;6.5 Exercises;150
12;Part III Stochastic Calculus for General Situations;153
13;7 G-Martingale Representation Theorem;154
13.1;7.1 G-Martingale Representation Theorem;154
14;8 Some Further Results of Itô's Calculus;164
14.1;8.1 A Generalized Itô's Integral;164
14.2;8.2 Itô's Integral for Locally Integrable Processes;170
14.3;8.3 Itô's Formula for General C2 Functions;174
15;Appendix A Preliminaries in Functional Analysis;178
16;A.1 Completion of Normed Linear Spaces;178
17;A.2 The Hahn-Banach Extension Theorem;179
18;A.3 Dini's Theorem and Tietze's Extension Theorem;179
19;Appendix B Preliminaries in Probability Theory;180
20;B.1 Kolmogorov's Extension Theorem;180
21;B.2 Kolmogorov's Criterion;181
22;B.3 Daniell-Stone Theorem;183
23;B.4 Some Important Inequalities;183
24;Appendix C Solutions of Parabolic Partial Differential Equation;185
25;C.1 The Definition of Viscosity Solutions;185
26;C.2 Comparison Theorem;187
27;C.3 Perron's Method and Existence;196
28;C.4 Krylov's Regularity Estimate for Parabolic PDEs;201
29;Appendix References;204
30;Appendix Index of Symbols;211
31;Appendix Author Index;213
32;Author Index;213
33;Appendix Subject Index;215
34;Index;215