Введение
1 Introduction 5
1.1 Aim and description of the work 5
1.2 Plan and main results 7
1.3 Conclusions and comments 10
2 Sublinear functionals and distributions 12
2.1 G-functional 12
2.2 Some remarks regarding the extension of the G-functional 19
2.3 Sublinear expectation 21
2.4 G-normal distribution 23
2.5 Covariance set under sublinear expectation 28
3 Viscosity solutions 34
3.1 B-continuity 34
3.2 Test functions and viscosity solutions 36
3.3 Comparison principle 38
3.4 Uniqueness of viscosity solution 41
4 G-expectations 43
4.1 G-Brownian motion 43
4.2 Capacity and upper expectation 45
4.3 Solving the fully nonlinear heat equation 46
4.4 Basic space constructions 49
4.5 Existence of G-normal distribution 55
4.6 Existence of G-Brownian motion and notion of G-expectation 58
4.7 G-expectation and upper expectation 64
5 Stochastic Integral with respect to G-Brownian motion 72
5.1 Definition of the stochastic integral 72
5.2 Ito’s isometry and Burkholder–Davis–Gundy inequalities . 76
5.3 Characterization of the space of integrand processes HMG2 . 84
5.4 Fubini theorem 87
5.5 Distribution of the stochastic integral 88
5.6 The continuity property of stochastic convolution 94
6 Viscosity solution for other parabolic PDEs 97
6.1 Ornstein-Uhlenbeck process 97
6.2 Solving the fully nonlinear parabolic PDE with a linear term 99
