In this paper we give some basic and important properties of several typical Banach spaces of functions of G-Brownian motion paths induced by a sublinear expectation-G-expectation. Many results can be also applied to more general situations. A generalized version of Kolmogorov's criterion for continuous modification of a stochastic process is also obtained. The results can be applied in continuous time dynamic and coherent risk measures in finance, in particular for path-dependence risky positions under situations of volatility model uncertainty.

We design a class of numerical schemes for backward stochastic differential equation driven by G-Brownian motion (G-BSDE), which is related to a fully nonlinear PDE. Based on Peng's central limit theorem, we employ the CLT method to approximate G-distributed. Rigorous stability and convergence analysis are also carried out. It is shown that the θ-scheme admits a half order convergence rate in the general case. In particular, for the case of θ 1 ∈ [0, 1] and θ 2 = 0, the scheme can reach first-order in the deterministic case. Several numerical tests are given to support our theoretical results.

In this paper, we study the recursive stochastic optimal control problems. The control domain does not need to be convex, and the generator of the backward stochastic differential equation can contain z. We obtain the variational equations for backward stochastic differential equations, and then obtain the maximum principle which solves completely Peng's open problem.

In this paper, we study comparison theorem, nonlinear Feynman-Kac formula and Girsanov transformation of the following BSDE driven by a G-Brownian motion. Yt = ξ + T t f (s, Ys, Zs)ds + T t g(s, Ys, Zs)d B s − T t ZsdBs − (KT − Kt), where K is a decreasing G-martingale.

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