Paper: arXiv 2308.12856

Authors: Marlon Moresco, Mélina Mailhot, Silvana M. Pesenti

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a sophisticated theoretical framework for dynamic robust risk measures using advanced probability theory, Wasserstein distances, and f-divergences. However, it is purely mathematical with no empirical backtesting, data validation, or quantitative experiments provided in the abstract.

Research Flowchart

  flowchart TD
    A[Goal: Quantify dynamic uncertainty propagation & evaluate robust risk measures] --> B[Input: Discrete Stochastic Processes over Finite Horizon]
    B --> C[Construct Dynamic Uncertainty Sets<br/>Wasserstein Distance & f-Divergences]
    C --> D[Define Dynamic Robust Risk Measures<br/>Supremum of risks within set]
    D --> E[Axiomatic Analysis<br/>Evaluate Convexity, Coherence & Time-Consistency]
    E --> F[Findings: f-divergences yield Strong Time-Consistency]
    E --> G[Findings: Wasserstein Distance yields Weak Recursiveness]
    F & G --> H[Outcome: Equivalent to recursive 1-step conditional measures]