容量限制理论和相关应用非线性数学期望（英文版）

　　《Capacity Limit Theory and Related Applications of Nonlinear Mathematical Expectation》内容介绍 ：Kolmogorov established the system of axioms for probability theory by Lebesgue’stheories of measure and integration in 1933, which make theory of probability to bethe important tool of investigating the random or uncertainty phenomena. However, ithas been shown that such additivity assumption of probabilities or linear expectationis not feasible in many areas of applications because the uncertainty and ambiguityphenomena, for example, Allais and Ellsberg paradoxes. The mathematical theory ofnon-additive measure and integral got its first important contribution with Choquet’sTheory of Capacities in 1954. Since then, capacities and Choquet integral are studiedby many researchers, for example, Huber and Strassen(1973), Walley and Fine (1982),Schmeidler(1989), Denneberg(1994), Maccheroni and Marinacci(2005), Chen (2010),and so on. Peng investigated the theory of nonlinear expectations from a new pointof view in 2006. This theory not based on probability space, but on nonlinear expec-tation space.

Chapter 1 Limit theory about capacity

1.1 Law of large numbers for capacity

1.1.1 Ambiguity urn models

1.1.2 Law of large numbers for BernouUi trials with ambiguity

1.1.3 General urn models

1.2 Weighted central limit theorem under sublinear expectations

1.2.1 Notations and preliminaries

1.2.2 Main result and proof

1.3 Berry-Esseen theory under linear expectation

1.4 Central limit theorem for capacity

Chapter 2 Discrete martingale under sublinear expectation

2.1 Definitions

2.2 SL-martingale and related inequalities

Chapter 3 Multi-dimensional G-Brownian motion

3.1 Kunita-Watanabe inequalities for multi-dimensional

G-Brownian motion

3.1.1 Preliminaries

3.1.2 Mutual variation process and Kunita-Watanabe inequalities for

multi-dimensional G-Brownian motion

3.2 Tanaka formula for multi-dimensional G-Brownian motion

Chapter 4 Stability problem for stochastic differential equations

driven by G-Brownian motion

4.1 Stability theorem for stochastic differential equations driven

by G-Brownian motion

4.1.1 Stability theorem for G-SDE under integral-Lipschitz condition

4.1.2 Stability about backward stochastic differential equations driven

by G-Brownian motion

4.1.3 Existence and uniqueness for forward-backward stochastic differential

equations driven by G-Brownian motion

4.1.4 Stability about forward-backward stochastic differential equations driven

by G-Brownian motion

4.2 Exponential stability for stochastic differential equations driven

by G-Brownian motion

4.2.1 Asymptotic Exponential stability for stochastic differential equations

driven by G-Brownian motion

4.3 Optimal control problems under G-expectation

4.3.1 Forward and backward stochastic differential equations driven

by G-Brownian motion

4.3.2 Optimal control problems under G-expectation

Chapter 5 Applications about G-Brownian motion in optimal

consumption and portfolio

5.1 Preliminaries

5.2 Optimal consumption and portfolio Rules under volatility uncertainty..

5.3 Mutual fund theorem under volatility uncertainty

5.4 A special case

Chapter 6 Functional solution about stochastic differential equation

driven by G-Brownian motion

6.1 Introduction

6.2 Functional solution about stochastic differential equation driven

by G-Brownian motion

6.3 Some classical models

6.3.1 Autonomous case

6.3.2 One-factor Hull-White model

6.3.3 Homogeneous linear G-stochastic differential equations

6.4 Conclusion

Bibliography

Symbol Index