  We study a robust utility maximization problem in a general discrete-time
frictionless market under quasi-sure no-arbitrage. The investor is assumed to
have a random and concave utility function defined on the whole real-line. She
also faces model ambiguity on her beliefs about the market, which is modeled
through a set of priors. We prove the existence of an optimal investment
strategy using only primal methods. For that we assume classical assumptions on
the market and on the random utility function as asymptotic elasticity
constraints. Most of our other assumptions are stated on a prior-by-prior basis
and correspond to generally accepted assumptions in the literature on markets
without ambiguity. We also propose a general setting including utility
functions with benchmark for which our assumptions are easily checked.
