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The target of this paper is to establish the bid-ask pricing frame work forthe American contingent claims against risky assets with G-asset price systems(see \cite{Chen2013b}) on the financial market under Knight uncertainty. First,we prove G-Dooby-Meyer decomposition for G-supermartingale. Furthermore, weconsider bid-ask pricing American contingent claims under Knight uncertain, byusing G-Dooby-Meyer decomposition, we construct dynamic superhedge stragies forthe optimal stopping problem, and prove that the value functions of the optimalstopping problems are the bid and ask prices of the American contingent claimsunder Knight uncertain. Finally, we consider a free boundary problem, prove thestrong solution existence of the free boundary problem, and derive that thevalue function of the optimal stopping problem is equivalent to the strongsolution to the free boundary problem.

G-Doob-Meyer Decomposition and its Application in Bid-Ask Pricing for American Contingent Claim Under Knightian Uncertainty