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Optimal Transport Divergences induced by Scoring Functions

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  • Silvana M. Pesenti
  • Steven Vanduffel

Abstract

We employ scoring functions, used in statistics for eliciting risk functionals, as cost functions in the Monge-Kantorovich (MK) optimal transport problem. This gives raise to a rich variety of novel asymmetric MK divergences, which subsume the family of Bregman-Wasserstein divergences. We show that for distributions on the real line, the comonotonic coupling is optimal for the majority of the new divergences. Specifically, we derive the optimal coupling of the MK divergences induced by functionals including the mean, generalised quantiles, expectiles, and shortfall measures. Furthermore, we show that while any elicitable law-invariant coherent risk measure gives raise to infinitely many MK divergences, the comonotonic coupling is simultaneously optimal. The novel MK divergences, which can be efficiently calculated, open an array of applications in robust stochastic optimisation. We derive sharp bounds on distortion risk measures under a Bregman-Wasserstein divergence constraint, and solve for cost-efficient payoffs under benchmark constraints.

Suggested Citation

  • Silvana M. Pesenti & Steven Vanduffel, 2023. "Optimal Transport Divergences induced by Scoring Functions," Papers 2311.12183, arXiv.org, revised Apr 2024.
    • Handle: RePEc:arx:papers:2311.12183
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    References listed on IDEAS

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    1. M. Burzoni & I. Peri & C. M. Ruffo, 2017. "On the properties of the Lambda value at risk: robustness, elicitability and consistency," Quantitative Finance, Taylor & Francis Journals, vol. 17(11), pages 1735-1743, November.
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    4. Gerber, Hans U., 1974. "On Additive Premium Calculation Principles," ASTIN Bulletin, Cambridge University Press, vol. 7(3), pages 215-222, March.
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    6. Chen, Zehua, 1996. "Conditional Lp-quantiles and their application to the testing of symmetry in non-parametric regression," Statistics & Probability Letters, Elsevier, vol. 29(2), pages 107-115, August.
    7. Fissler, Tobias & Pesenti, Silvana M., 2023. "Sensitivity measures based on scoring functions," European Journal of Operational Research, Elsevier, vol. 307(3), pages 1408-1423.
    8. Fabio Bellini & Valeria Bignozzi, 2015. "On elicitable risk measures," Quantitative Finance, Taylor & Francis Journals, vol. 15(5), pages 725-733, May.
    9. Carole Bernard & Silvana M. Pesenti & Steven Vanduffel, 2024. "Robust distortion risk measures," Mathematical Finance, Wiley Blackwell, vol. 34(3), pages 774-818, July.
    10. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
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    Cited by:

    1. Xia Han & Peng Liu, 2024. "Robust Lambda-quantiles and extreme probabilities," Papers 2406.13539, arXiv.org.

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