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Distributional Transforms, Probability Distortions, and Their Applications

Author

Listed:
  • Peng Liu

    (Department of Mathematical Sciences, University of Essex, Colchester, CO4 3SQ, United Kingdom)

  • Alexander Schied

    (Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

  • Ruodu Wang

    (Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

Abstract

In this paper we provide a general mathematical framework for distributional transforms, which allows for many examples that are used extensively in the literature of finance, economics, and optimization. We put a special focus on the class of probability distortions, which is a fundamental tool in decision theory. As our main results, we characterize distributional transforms satisfying various properties, and this includes an equivalent set of conditions which forces a distributional transform to be a probability distortion. As the first application, we construct new risk measures using distributional transforms. Sufficient and necessary conditions are given to ensure the convexity or coherence of the generated risk measures. In the second application, we introduce a new method for sensitivity analysis of risk measures based on composition groups of probability distortions. Finally, we construct probability distortions describing a change of measures with an example in option pricing.

Suggested Citation

  • Peng Liu & Alexander Schied & Ruodu Wang, 2021. "Distributional Transforms, Probability Distortions, and Their Applications," Mathematics of Operations Research, INFORMS, vol. 46(4), pages 1490-1512, November.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:4:p:1490-1512
    DOI: 10.1287/moor.2020.1090
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    References listed on IDEAS

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    Cited by:

    1. Aurélien Baillon & Han Bleichrodt & Chen Li & Peter P. Wakker, 2025. "Source Theory: A Tractable and Positive Ambiguity Theory," Management Science, INFORMS, vol. 71(10), pages 8767-8782, October.
    2. Christopher P. Chambers & Peng Liu & Ruodu Wang, 2025. "A duality between utility transforms and probability distortions," Theory and Decision, Springer, vol. 99(4), pages 757-779, December.
    3. Ruodu Wang & Qinyu Wu, 2025. "Probabilistic risk aversion for generalized rank-dependent functions," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 79(3), pages 1055-1082, May.

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