Finance and Stochastics

, Volume 23, Issue 1, pp 209–238 | Cite as

Minimax theorems for American options without time-consistency

  • Denis Belomestny
  • Tobias Hübner
  • Volker KrätschmerEmail author
  • Sascha Nolte


In this paper, we give sufficient conditions guaranteeing the validity of the well-known minimax theorem for the lower Snell envelope. Such minimax results play an important role in the characterisation of arbitrage-free prices of American contingent claims in incomplete markets. Our conditions do not rely on the notions of stability under pasting or time-consistency and reveal some unexpected connection between the minimax result and path properties of the corresponding process of densities. We exemplify our general results in the case of families of measures corresponding to diffusion exponential martingales.


Minimax Lower Snell envelope Time-consistency Nearly sub-Gaussian random fields Metric entropies Simons’ lemma 

Mathematics Subject Classification (2010)

60G40 90C47 91G20 60G17 

JEL Classification

C73 G12 D81 



The authors would like to thank Mikhail Urusov for fruitful discussions and helpful remarks. We are also grateful to some anonymous referee and the Co-Editor Alexander Schied for valuable suggestions to improve the presentation.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Denis Belomestny
    • 1
    • 2
  • Tobias Hübner
    • 1
  • Volker Krätschmer
    • 1
    Email author
  • Sascha Nolte
    • 1
  1. 1.Faculty of MathematicsUniversity of Duisburg–EssenEssenGermany
  2. 2.National University Higher School of EconomicsMoscowRussia

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