Abstract
We unify and establish equivalence between the pathwise and the quasi-sure approaches to robust modelling of financial markets in discrete time. In particular, we prove a fundamental theorem of asset pricing and a superhedging theorem, which encompass the formulations of Bouchard and Nutz (Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25:823-859, 2015) and Burzoni et al. (Pointwise arbitrage pricing theory in discrete time, Math. Oper. Res. 43:1034-1057, 2019). In bringing the two streams of literature together, we examine and compare their many different notions of arbitrage. We also clarify the relation between robust and classical P-specific results. Furthermore, we prove when a superhedging property with respect to the set of martingale measures supported on a set of paths Ω may be extended to a pathwise superhedging on Ω without changing the superhedging price.