Paper: arXiv 2504.00158

Authors: Alexandre Boistard, Laurence Carassus, Safae Issaoui

Abstract

Drawing on set theory, this paper contributes to a deeper understanding of the structural condition of mathematical finance under Knightian uncertainty. We adopt a projective framework in which all components of the model – prices, priors and trading strategies – are treated uniformly in terms of measurability. This contrasts with the quasi-sure setting of Bouchard and Nutz, in which prices are Borel-measurable and graphs of local priors are analytic sets, while strategies and stochastic kernels inherit only universal measurability. In our projective framework, we establish several characterizations of the robust no-arbitrage condition, already known in the quasi-sure setting, but under significantly more elegant and consistent assumptions. These characterisations have important applications, in particular, the existence of solutions to the robust utility maximization problem. To do this, we work within the classical Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), augmented by the axiom of Projective Determinacy (PD). The (PD) axiom, a well-established axiom of descriptive set theory, guarantees strong regularity properties for projective sets and projective functions.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper offers a highly advanced theoretical contribution to quantitative finance by applying descriptive set theory and the Axiom of Projective Determinacy to robust no-arbitrage theory. While mathematically rigorous and resolving open theoretical conjectures, it is strictly pure mathematics without empirical backtesting or real-world market data.

Research Flowchart

  flowchart TD
    A[Goal: Unify measurability in robust math finance under Knightian uncertainty] --> B[Foundational Inputs: ZFC Set Theory + Projective Determinacy PD Axiom]
    B --> C[Methodology: Adopt Projective Framework for prices, priors, and strategies]
    C --> D[Analytical Process: Contrast with quasi-sure setting to refine measurability assumptions]
    D --> E[Outcome 1: Uniform characterization of Robust No-Arbitrage condition]
    D --> F[Outcome 2: Proof of existence for robust utility maximization solutions]