G-consistent price system and bid-ask pricing for European contingent claims under Knightian uncertainty

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.