Nonlinear expectations and stochastic calculus under uncertainty

✍ Scribed by Peng S

• Publisher: Springer
• Year: 2019
• Tongue: English
• Leaves: 216
• Series: PTSM 95
• Category: Library

✦ Table of Contents

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