  The target of this paper is to consider model the risky asset price on the
financial market under the Knightian uncertainty, and pricing the ask and bid
prices of the uncertain risk. We use the nonlinear analysis tool, i.e., G-frame
work [26], to construct the model of the risky asset price and bid-ask pricing
for the European contingent claims under Knightian uncertain financial market.
Firstly, we consider the basic risky asset price model on the uncertain
financial market, which we construct here is the model with drift uncertain and
volatility uncertain. We describe such model by using generalized G-Brownian
motion and call it as G-asset price system. We present the uncertain risk
premium which is uncertain and distributed with maximum distribution. We derive
the closed form of bid-ask price of the European contingent claim against the
underlying risky asset with G-asset price system as the discounted conditional
G-expecation of the claim, and the bid and ask prices are the viscosity
solutions to the nonlinear HJB equations.Furthermore, we consider the main part
of this paper, i.e., consider the risky asset on the Knightian uncertain
financial market with the price fluctuation shows as continuous trajectories.
We propose the G-conditional full support condition by using uncertain
capacity, and the risky asset price path satisfying the G-conditional full
support condition could be approximated by its G-consistent asset price
systems. We derive that the bid and ask prices of the European contingent claim
against such risky asset under uncertain can be expressed by discounted of some
conditional G-expectation of the claim. We give examples, such as G-Markovian
processes and the geometric fractional G-Brownian motion [9], satisfying the
G-conditional full support condition.
