P-Sensitive Functions and Localizations

TL;DR

The paper develops a rigorous framework for robust analysis under Knightian uncertainty by introducing $\mathcal{P}$-sensitive functions on the robust space $L^{0}_{c}$ and proving that such functions admit a representation through functional localization. It establishes a precise link between $\mathcal{P}$-sensitivity and both primal and dual localizations, enabling decomposition of complex robust problems into tractable local (dominated) problems and a controlled aggregation. The authors apply this to robust optimization, convex risk measures, and one-period arbitrage/superhedging, deriving localization bubbles, dual representations, and robust FTAP results under various conditions. The work provides a unified method to transfer classical dominated-model techniques to non-dominated settings and clarifies when localized representations yield consistent pricing or risk assessments across models. These results have practical implications for risk management and pricing in uncertain markets where model ambiguity cannot be captured by a single dominating measure.

Abstract

This paper assumes a robust stochastic model where a set $\mathcal{P}$ of probability measures replaces the single probability measure of dominated models. We introduce and study $\mathcal{P}$-sensitive functions defined on robust function spaces of random variables. We show that $\mathcal{P}$-sensitive functions are precisely those that admit a representation via so-called functional localization. The theory is applied to solving robust optimization problems, to convex risk measures, and to the study of no arbitrage in robust one-period financial models.

Theorems & Definitions (107)

• Definition 2.1
• Definition 2.2
• Lemma 2.3: LS2025
• Lemma 2.4: LS2025
• Lemma 2.5
• proof
• Definition 2.6
• Definition 2.7
• Lemma 2.8
• proof