Uncertainty Propagation and Dynamic Robust Risk Measures

TL;DR

This paper develops a framework for modeling how uncertainty propagates over time in stochastic processes, introducing dynamic uncertainty sets and robust risk measures that account for distributional ambiguity and ensure desirable properties like time-consistency.

Contribution

It introduces a novel framework for dynamic uncertainty sets and robust risk measures, analyzing their properties and conditions for time-consistency in stochastic processes.

Findings

Uncertainty sets from f-divergences lead to strong time-consistency.

Wasserstein-based uncertainty sets result in a new form of weak recursiveness.

Dynamic robust risk measures admit recursive representations under certain conditions.

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.