Equivalence between time consistency and nested formula

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.