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Optimal Stopping under Adverse Nonlinear Expectation and Related Games

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  • Marcel Nutz
  • Jianfeng Zhang
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    Abstract

    We study the existence of optimal actions in a zero-sum game $\inf_\tau \sup_P E^P[X_\tau]$ between a stopper and a controller choosing a probability measure. In particular, we consider the optimal stopping problem $\inf_\tau \mathcal{E}(X_\tau)$ for a class of sublinear expectations $\mathcal{E}(\cdot)$ including the $G$-expectation. We show that the game has a value. Moreover, exploiting the theory of sublinear expectations, we define a nonlinear Snell envelope $Y$ and prove that the first hitting time $\inf{t:\, Y_t=X_t}$ is an optimal stopping time. The existence of a saddle point is shown under a compactness condition. Finally, the results are applied to the subhedging of American options under volatility uncertainty.

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    File URL: http://arxiv.org/pdf/1212.2140
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1212.2140.

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    Date of creation: Dec 2012
    Date of revision: Jul 2014
    Handle: RePEc:arx:papers:1212.2140

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    Web page: http://arxiv.org/

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    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    1. Frank Riedel, 2009. "Optimal Stopping With Multiple Priors," Econometrica, Econometric Society, Econometric Society, vol. 77(3), pages 857-908, 05.
    2. Ariel Neufeld & Marcel Nutz, 2012. "Superreplication under Volatility Uncertainty for Measurable Claims," Papers, arXiv.org 1208.6486, arXiv.org, revised Apr 2013.
    3. Marcel Nutz & Ramon van Handel, 2012. "Constructing Sublinear Expectations on Path Space," Papers, arXiv.org 1205.2415, arXiv.org, revised Apr 2013.
    4. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 2(2), pages 117-133.
    5. Adam Smith, 2002. "American options under uncertain volatility," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 9(2), pages 123-141.
    6. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 2(2), pages 73-88.
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    Cited by:
    1. Erhan Bayraktar & Song Yao, 2013. "On the Robust Optimal Stopping Problem," Papers, arXiv.org 1301.0091, arXiv.org, revised Jul 2014.
    2. Bruno Bouchard & Marcel Nutz, 2013. "Arbitrage and Duality in Nondominated Discrete-Time Models," Papers, arXiv.org 1305.6008, arXiv.org, revised Feb 2014.
    3. Marcel Nutz, 2014. "Robust Superhedging with Jumps and Diffusion," Papers, arXiv.org 1407.1674, arXiv.org.

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