Uncertainty Propagation and Dynamic Robust Risk Measures

32 Pages Posted: 27 Aug 2023 Last revised: 6 Feb 2024

See all articles by Marlon Moresco

Marlon Moresco

Concordia University

Mélina Mailhot

Concordia University

Silvana M. Pesenti

University of Toronto

Date Written: August 24, 2023

Abstract

We introduce a framework for quantifying propagation of uncertainty arising in a dynamic setting. Specifically, we define dynamic uncertainty sets designed explicitly for discrete stochastic processes over a finite time horizon. These dynamic uncertainty sets capture the uncertainty surrounding stochastic processes and models, accounting for factors such as distributional ambiguity. Examples of uncertainty sets include those induced by the Wasserstein distance and f-divergences.

We further define dynamic robust risk measures as the supremum of all candidates' risks within the uncertainty set. In an axiomatic way, we discuss conditions on the uncertainty sets that lead to well-known properties of dynamic robust risk measures, such as convexity and coherence. Furthermore, we discuss the necessary and sufficient properties of dynamic uncertainty sets that lead to time-consistencies of dynamic robust risk measures. We find that uncertainty sets stemming from f-divergences lead to strong time-consistency while the Wasserstein distance results in a new time-consistent notion of weak recursiveness. Moreover, we show that a dynamic robust risk measure is strong time-consistent or weak recursive if and only if it admits a recursive representation of one-step conditional robust risk measures arising from static uncertainty sets.

Keywords: Dynamic Risk Measures, Time-consistency, Distributional Uncertainty, Wasserstein distance

Suggested Citation

Moresco, Marlon and Mailhot, Mélina and Pesenti, Silvana M., Uncertainty Propagation and Dynamic Robust Risk Measures (August 24, 2023). Available at SSRN: https://ssrn.com/abstract=4551069 or http://dx.doi.org/10.2139/ssrn.4551069

Marlon Moresco (Contact Author)

Concordia University ( email )

Canada

Mélina Mailhot

Concordia University

Silvana M. Pesenti

University of Toronto ( email )

700 University Avenue 9F
Toronto, Ontario
Canada

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