Robust No-Arbitrage under Projective Determinacy

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.