Nonlinear Expectations and Stochastic Calculus under Uncertainty (eBook)

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