G-Doob-Meyer Decomposition and its Application in Bid-Ask Pricing for American Contingent Claim Under Knightian Uncertainty
Wei Chen
TL;DR
This work develops a rigorous bid-ask pricing framework for American contingent claims under Knightian uncertainty using G-asset price systems. It proves a G-Doob-Meyer decomposition for G-supermartingales, constructs dynamic superhedging strategies through optimal stopping, and shows that the resulting value functions serve as the ask and bid prices under Knight uncertainty. The analysis links the stopping problem to a free boundary PDE with the operator $\\mathcal{L}u=G(D^2u)+rDu+\partial_tu$, proving existence and regularity of a strong solution and establishing that this solution equals the stopping-value. Collectively, the results provide a principled approach to pricing and hedging American options in incomplete markets with volatility and drift ambiguity, under a sublinear expectation framework.
Abstract
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.