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Equivalence Between Time Consistency and Nested Formula


  • Henri G'erard


  • Michel de Lara


  • Jean-Philippe Chancelier



You are a financial analyst. At the beginning of every week, you are able to rank every pair of stochastic processes starting from that week up to the horizon. Suppose that two processes are equal at the beginning of the week. Your ranking procedure is time consistent if the ranking does not change between this week and the next one. In this paper, we propose a minimalist definition of Time Consistency (TC) between two (assessment) mappings. With very few assumptions, we are able to prove an equivalence between Time Consistency and a Nested Formula (NF) between the two mappings. Thus, in a sense, two assessments are consistent if and only if one is factored into the other. We review the literature and observe that the various definitions of TC (or of NF) are special cases of ours, as they always include additional assumptions. By stripping off these additional assumptions, we present an overview of the literature where the contribution of each author is enlightened.

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  • Henri G'erard & Michel de Lara & Jean-Philippe Chancelier, 2017. "Equivalence Between Time Consistency and Nested Formula," Papers 1711.08633,, revised May 2019.
  • Handle: RePEc:arx:papers:1711.08633

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    References listed on IDEAS

    1. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
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    6. De Lara, Michel & Leclère, Vincent, 2016. "Building up time-consistency for risk measures and dynamic optimization," European Journal of Operational Research, Elsevier, vol. 249(1), pages 177-187.
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    8. Peter J. Hammond, 1976. "Changing Tastes and Coherent Dynamic Choice," Review of Economic Studies, Oxford University Press, vol. 43(1), pages 159-173.
    9. Kreps, David M & Porteus, Evan L, 1978. "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica, Econometric Society, vol. 46(1), pages 185-200, January.
    10. Alexander Shapiro, 2016. "Rectangular Sets of Probability Measures," Operations Research, INFORMS, vol. 64(2), pages 528-541, April.
    11. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath & Hyejin Ku, 2007. "Coherent multiperiod risk adjusted values and Bellman’s principle," Annals of Operations Research, Springer, vol. 152(1), pages 5-22, July.
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