Equivalence Between Time Consistency and Nested Formula

TL;DR

This paper establishes a fundamental equivalence between time consistency and a nested formula in the context of stochastic process ranking, providing a minimalist framework that unifies various existing definitions.

Contribution

It introduces a minimalistic definition of time consistency and proves its equivalence to a nested formula, unifying and generalizing previous approaches.

Findings

Proves equivalence between time consistency and nested formula

Unifies various existing definitions under a minimal framework

Provides a comprehensive literature review highlighting assumptions

Abstract

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.