Equivalence between time consistency and nested formula
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Equivalence between time consistency and nested formula Henri Gérard1,2
· Michel De Lara1 · Jean-Philippe Chancelier1
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Figure out a situation where, at the beginning of every week, one has 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. The 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 specific contributions of authors are enlightened. Moreover, we present two classes of mappings, translation invariant mappings and Fenchel–Moreau conjugates, that display time consistency under suitable assumptions. Keywords Dynamic risk measure · Time consistency · Nested formula
1 Introduction Behind the words “time consistency” and “nested formula”, one can find a vast literature resorting to economics, dynamical risk measures and stochastic optimization. Let us start with economics. In a dynamic bargaining problem, a group of agents has to agree on a common path of actions. As time goes on and information is progressively revealed, they can all reconsider the past agreement, and possibly make new assessments leading to
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Henri Gérard [email protected] Michel De Lara [email protected] Jean-Philippe Chancelier [email protected]
1
Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée, France
2
Université Paris-Est, Labex Bézout, 77455 Marne-la-Vallée, France
123
Annals of Operations Research
new actions. Stability is the property that the agents will stick to their previous commitment. Time consistency is a form of stability when an individual makes a deal between his different selves (agents) along time. The notion of “consistent course of action” (see Peleg and Yaari 1973) is well-known in the field of economics, with the seminal work of Strotz (1955, 1956): an individual having planned his consumption trajectory is consistent if, reevaluating his plans later on, he does not deviate from the originally chosen plan. This idea of consistency as “sticking to one’s plan” may be extended to the uncertain case where plans are replaced by decision rules [“Do thus-and-thus if you find yourself in this portion of state space with this amount of time left”, Richard Bellman cited in Dreyfus (2002)]; Hammond (1976) addresses “consistency” and “coherent dynamic choice”, Kreps
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