  G-framework is presented by Peng [41] for measure risk under uncertainty. In
this paper, we define fractional G-Brownian motion (fGBm). Fractional
G-Brownian motion is a centered G-Gaussian process with zero mean and
stationary increments in the sense of sub-linearity with Hurst index $H\in
(0,1)$. This process has stationary increments, self-similarity, and long rang
dependence properties in the sense of sub-linearity. These properties make the
fractional G-Brownian motion a suitable driven process in mathematical finance.
We construct wavelet decomposition of the fGBm by wavelet with compactly
support. We develop fractional G-white noise theory, define G-It\^o-Wick
stochastic integral, establish the fractional G-It\^o formula and the
fractional G-Clark-Ocone formula, and derive the G-Girsanov's Theorem. For
application the G-white noise theory, we consider the financial market modelled
by G-Wick-It\^o type of SDE driven by fGBm. The financial asset price modelled
by fGBm has volatility uncertainty, using G-Girsanov's Theorem and
G-Clark-Ocone Theorem, we derive that sublinear expectation of the discounted
European contingent claim is the bid-ask price of the claim.
