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Inequalities for upper and lower probabilities

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  • Chen, Zengjing
  • Kulperger, Reg

Abstract

An upper (lower) probability can be defined as a supremum (infimum) over a set of probability measures. The upper probability measure is also called a capacity. A capacity is usually studied under an assumption that it is 2-alternating, but many capacities are not 2-alternating. In this paper, we introduce a new definition of sub 2-alternating, defined for a capacity and its conjugate capacity. We study these inequalities for a certain natural capacity that is the supremum over a particular set of probability measures with a particular form of Radon Nikodym derivative with respect to Brownian motion. This capacity is not 2-alternating, but the pair of the capacity and its conjugate capacity is sub 2-alternating.

Suggested Citation

  • Chen, Zengjing & Kulperger, Reg, 2005. "Inequalities for upper and lower probabilities," Statistics & Probability Letters, Elsevier, vol. 73(3), pages 233-241, July.
    • Handle: RePEc:eee:stapro:v:73:y:2005:i:3:p:233-241
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    References listed on IDEAS

    as
    1. Chen, Zengjing & Kulperger, Reg & Wei, Gang, 2005. "A comonotonic theorem for BSDEs," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 41-54, January.
    2. Zengjing Chen & Larry Epstein, 2002. "Ambiguity, Risk, and Asset Returns in Continuous Time," Econometrica, Econometric Society, vol. 70(4), pages 1403-1443, July.
    3. 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.
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