Optimal stopping under adverse nonlinear expectation and related games

Author Info

• Marcel Nutz
• Jianfeng Zhang
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Abstract

We study the existence of optimal actions in a zero-sum game $\inf_{\tau}\sup_PE^P[X_{\tau}]$ between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem $\inf_{\tau}\mathcal{E}(X_{\tau})$ for a class of sublinear expectations $\mathcal{E}(\cdot)$ such as 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

Bibliographic Info

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

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 Length: Date of creation: Dec 2012 Date of revision: Sep 2015 Publication status: Published in Annals of Applied Probability 2015, Vol. 25, No. 5, 2503-2534 Handle: RePEc:arx:papers:1212.2140 Contact details of provider: Web page: http://arxiv.org/

References

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1. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
2. Frank Riedel, 2009. "Optimal Stopping With Multiple Priors," Econometrica, Econometric Society, vol. 77(3), pages 857-908, 05.
3. Erhan Bayraktar & Yu-Jui Huang, 2010. "On the Multi-Dimensional Controller and Stopper Games," Papers 1009.0932, arXiv.org, revised Jan 2013.
4. Marcel Nutz & Ramon van Handel, 2012. "Constructing Sublinear Expectations on Path Space," Papers 1205.2415, arXiv.org, revised Apr 2013.
5. Erhan Bayraktar & Ioannis Karatzas & Song Yao, 2009. "Optimal Stopping for Dynamic Convex Risk Measures," Papers 0909.4948, arXiv.org, revised Nov 2009.
6. Adam Smith, 2002. "American options under uncertain volatility," Applied Mathematical Finance, Taylor & Francis Journals, vol. 9(2), pages 123-141.
7. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
8. Ariel Neufeld & Marcel Nutz, 2012. "Superreplication under Volatility Uncertainty for Measurable Claims," Papers 1208.6486, arXiv.org, revised Apr 2013.
9. Erhan Bayraktar & Song Yao, 2009. "Optimal Stopping for Non-linear Expectations," Papers 0905.3601, arXiv.org, revised Jan 2011.
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