Muckenhoupt's $(A_p)$ condition and the existence of the optimal martingale measure

Abstract

In the problem of optimal investment with utility function defined on $(0,\infty )$, 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 $(A_p)$ condition for the power $p=1/(1-a)$, where $a\in (0,1)$ is a lower bound on the relative risk-aversion of the utility function. We construct a counterexample showing that this $(A_p)$ condition is sharp.

24 pages

References in corpus (1)

• On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets

Cited by in corpus (1)

• On the dual problem of utility maximization in incomplete markets