The von Neumann–Morgenstern rationality axioms and analytic inequalities

Trading in an efficient market where the asset price behaves as a martingale leads to a zero expected payoff. However, the problem of how to make such trading as rational as possible remains meaningful and non-trivial: it turns out that there is a certain gap between trading that has zero profit expectation, but is still rational in the basic sense, and completely irrational economic behavior that violates the basic von Neumann–Morgenshtern rationality axioms. By solving the problem of describing this gap and finding optimal trading strategies that get into it, we will arrive at the Bellman functions that have previously arisen in solving completely abstract problems about finding sharp constants in inequalities from analysis. Namely, solving the economic problem in the absolute context, where the strategy to be chosen does not depend on the current wealth of the agent, we will arrive at the Bellman function related to the John–Nirenberg inequality in integral form. Solving the problem in a relative context, where all the agent's actions in the market are considered relative to his current wealth, we will arrive at the Bellman function related to the inequalities that describe the relationship between Gehring classes. Thus, we will obtain a natural economic interpretation for the listed inequalities and the Bellman functions associated with them.