Everywhere Differentiable Continuums, Mathematical Spaces, and Axioms: Modeling of Financial Phenomena

Formal theoretical proofs show modeling of stock returns on `everywhere differentiable' continuums always is inappropriate to modeling of rational expectations equilibriums (REE). Simultaneously, modeling of stock returns in discrete time always is robust to modeling of each of REE, or feasibility of deviations from REE. In stated respect, modeling in discrete time results in dichotomous sufficiency conditions for each of conformance with, or deviation of investors' priors from REE. Bane of modeling in continuous time is non-dichotomization of connectedness property of stock prices from evolution of stock prices, a contradiction to the norm that connectedness properties be independent of specific elements that are located in topological spaces. In aggregate, while modeling in continuous time induces positive relations between risk (volatility) and returns, contrary to rational expectations, it is relatively low realizations for volatility that have higher risk of generation of negative returns.