Muckenhoupt’s (Ap)(Ap) condition and the existence of the optimal martingale measure

In the problem of optimal investment with a utility function defined on (0,∞)(0,∞), we formulate sufficient conditions for the dual optimizer to be a uniformly integrable martingale. Our key requirement consists of the existence of a martingale measure whose density process satisfies the probabilistic Muckenhoupt (Ap)(Ap) condition for the power p=1/(1−a)p=1/(1−a), where a∈(0,1)a∈(0,1) is a lower bound on the relative risk-aversion of the utility function. We construct a counterexample showing that this (Ap)(Ap) condition is sharp.