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

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  • Wei Chen

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.

Suggested Citation

  • Wei Chen, 2013. "G-Doob-Meyer Decomposition and its Application in Bid-Ask Pricing for American Contingent Claim Under Knightian Uncertainty," Papers 1401.0677, arXiv.org.
  • Handle: RePEc:arx:papers:1401.0677
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    File URL: http://arxiv.org/pdf/1401.0677
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    References listed on IDEAS

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    1. Kramkov, D.O., 1994. "Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets," Discussion Paper Serie B 294, University of Bonn, Germany.
    2. Wei Chen, 2013. "Fractional G-White Noise Theory, Wavelet Decomposition for Fractional G-Brownian Motion, and Bid-Ask Pricing Application to Finance Under Uncertainty," Papers 1306.4070, arXiv.org.
    3. RĂ˜diger Frey, 2000. "Superreplication in stochastic volatility models and optimal stopping," Finance and Stochastics, Springer, vol. 4(2), pages 161-187.
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    5. Wei Chen, 2013. "G-consistent price system and bid-ask pricing for European contingent claims under Knightian uncertainty," Papers 1308.6256, arXiv.org, revised Sep 2013.
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