Fundamental Theorem of Asset Pricing on Measurable Spaces Under Uncertainty

Abstract

It is common in the financial mathematics literature to start by fixing a probability space $(\Omega,\mathcal F,\mathbb P)$, on which the underlying price process is defined. We depart from this route in that we do not fix the prior $\mathbb P$. Under very general assumptions, we recover the Fundamental Theorem of Asset Pricing in discrete time under either a multiple-priors or a prior-free setting. We only require that $(\Omega, \mathcal F)$ is a measurable space, while the multiple priors can be non-equivalent. Furthermore, the initial price of our market model does not need to be constant, but only measurable.

Keywords: Fundamental Theorem of Asset Pricing, uncertainty, multiple priors

JEL Classification: C00, G12