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Root’s barrier, viscosity solutions of obstacle problems and reflected FBSDEs

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  • Gassiat, Paul
  • Oberhauser, Harald
  • dos Reis, Gonçalo

Abstract

We revisit work of Rost (1976), Dupire (2005) and Cox–Wang (2013) on connections between Root’s solution of the Skorokhod embedding problem and obstacle problems. We develop an approach based on viscosity sub- and supersolutions and an accompanying comparison principle. This gives new, constructive and simple proofs of the existence and minimality properties of Root type solutions as well as their complete characterization. The approach is self-contained and covers martingale diffusions with degenerate elliptic or time-dependent volatility as well as Rost’s reversed Root barriers; it also provides insights about the dynamics of general Skorokhod embeddings by identifying them as supersolutions of certain nonlinear PDEs.

Suggested Citation

  • Gassiat, Paul & Oberhauser, Harald & dos Reis, Gonçalo, 2015. "Root’s barrier, viscosity solutions of obstacle problems and reflected FBSDEs," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4601-4631.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:12:p:4601-4631
    DOI: 10.1016/j.spa.2015.07.010
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    References listed on IDEAS

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    1. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2013. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Sciences Po publications info:hdl:2441/5rkqqmvrn4t, Sciences Po.
    2. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    3. Alexander M. G. Cox & Jiajie Wang, 2013. "Optimal robust bounds for variance options," Papers 1308.4363, arXiv.org.
    4. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    5. Alexander M. G. Cox & Jiajie Wang, 2011. "Root's barrier: Construction, optimality and applications to variance options," Papers 1104.3583, arXiv.org, revised Mar 2013.
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    Cited by:

    1. Guo, Gaoyue & Tan, Xiaolu & Touzi, Nizar, 2017. "Tightness and duality of martingale transport on the Skorokhod space," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 927-956.
    2. Henry-Labordère, Pierre & Tan, Xiaolu & Touzi, Nizar, 2016. "An explicit martingale version of the one-dimensional Brenier’s Theorem with full marginals constraint," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2800-2834.
    3. Pierre Henry-Labordère & Nizar Touzi, 2016. "An explicit martingale version of the one-dimensional Brenier theorem," Finance and Stochastics, Springer, vol. 20(3), pages 635-668, July.
    4. Wang, Jiajie, 2020. "Minimal Root’s embeddings for general starting and target distributions," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 521-544.
    5. Mathias Beiglboeck & Alexander Cox & Martin Huesmann, 2017. "The geometry of multi-marginal Skorokhod Embedding," Papers 1705.09505, arXiv.org.
    6. De Angelis, Tiziano & Kitapbayev, Yerkin, 2017. "Integral equations for Rost’s reversed barriers: Existence and uniqueness results," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3447-3464.

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