目录
- Contents
Chapter 1 Limit theory about capacity . . 1
1.1 Law of large numbers for capacity . . . . . . 1
1.1.1 Ambiguity urn models . . . 1
1.1.2 Law of large numbers for Bernoulli trials with ambiguity . 3
1.1.3 General urn models . . . . . 11
1.2 Weighted central limit theorem under sublinear expectations . . 13
1.2.1 Notations and preliminaries . . . . . . . 14
1.2.2 Main result and proof . . .16
1.3 Berry-Esseen theory under linear expectation. . .24
1.4 Central limit theorem for capacity. . . . .27
Chapter 2 Discrete martingale under sublinear expectation. . 30
2.1 Definitions . 30
2.2 SL-martingale and related inequalities . . 33
Chapter 3 Multi-dimensional G-Brownian motion. . . . . . 41
3.1 Kunita-Watanabe inequalities for multi-dimensional G-Brownian motion . . 41
3.1.1 Preliminaries . . . . 41
3.1.2 Mutual variation process and Kunita-Watanabe inequalities for multi-dimensional G-Brownian motion . . . . .44
3.2 Tanaka formula for multi-dimensional G-Brownian motion. . . . .52
Chapter 4 Stability problem for stochastic differential equations driven by G-Brownian motion . 56
4.1 Stability theorem for stochastic differential equations driven by G-Brownian motion . . . . . 56
4.1.1 Stability theorem for G-SDE under integral-Lipschitz condition . .57
4.1.2 Stability about backward stochastic differential equations driven by G-Brownian motion . . 60
4.1.3 Existence and uniqueness for forward-backward stochastic differential equations driven by G-Brownian motion . . . 63
4.1.4 Stability about forward-backward stochastic differential equations driven by G-Brownian motion . . 66
vi Contents
4.2 Exponential stability for stochastic differential equations driven by G-Brownian motion . 66
4.2.1 Asymptotic Exponential stability for stochastic differential equations driven by G-Brownian motion . . . . . 68
4.3 Optimal control problems under G-expectation . 76
4.3.1 Forward and backward stochastic differential equations driven by G-Brownian motion . . 76
4.3.2 Optimal control problems under G-expectation. . . . . . .79
Chapter 5 Applications about G-Brownian motion in optimal consumption and portfolio . . . 87
5.1 Preliminaries. .87
5.2 Optimal consumption and portfolio Rules under volatility uncertainty . . . 88
5.3 Mutual fund theorem under volatility uncertainty . . . . . . 97
5.4 A special case. . . . . . .100
Chapter 6 Functional solution about stochastic differential equation driven by G-Brownian motion. . . . .104
6.1 Introduction . 104
6.2 Functional solution about stochastic differential equation driven by G-Brownian motion . . . . . 105
6.3 Some classical models . 109
6.3.1 Autonomous case . . . . . .109
6.3.2 One-factor Hull-White model . . . . . 111
6.3.3 Homogeneous linear G-stochastic differential equations . . . . . 113
6.4 Conclusion. .115
Bibliography . . . . 116
Symbol Index . . . 123
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