https://hal-enpc.archives-ouvertes.fr/hal-01645564v2Gérard, HenriHenriGérardCERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTechde Lara, MichelMichelde LaraCERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTechChancelier, Jean-PhilippeJean-PhilippeChancelierCERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTechMATHRISK - Mathematical Risk Handling - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École des Ponts ParisTech - Inria de Paris - Inria - Institut National de Recherche en Informatique et en AutomatiqueEquivalence Between Time Consistency and Nested FormulaHAL CCSD2019Nested FormulaDynamic Risk MeasureTime Consistency[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]Gérard, Henri2019-05-17 14:13:342022-07-08 10:06:152019-05-17 16:18:01enJournal articleshttps://hal-enpc.archives-ouvertes.fr/hal-01645564v2/document10.1007/s10479-019-03276-1https://hal-enpc.archives-ouvertes.fr/hal-01645564v1application/pdf2You 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.