  The target of this paper is to establish the bid-ask pricing frame work for
the 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, we
consider bid-ask pricing American contingent claims under Knight uncertain, by
using G-Dooby-Meyer decomposition, we construct dynamic superhedge stragies for
the optimal stopping problem, and prove that the value functions of the optimal
stopping problems are the bid and ask prices of the American contingent claims
under Knight uncertain. Finally, we consider a free boundary problem, prove the
strong solution existence of the free boundary problem, and derive that the
value function of the optimal stopping problem is equivalent to the strong
solution to the free boundary problem.

Probability;Pricing of Securities