A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: \begin{myenumerate} \item The optimal terminal wealth $X^*(T) : = X_{\varphi^*}(T)$ of the classical problem to maximize the expected $U$-utility of the terminal wealth $X_{\varphi}(T)$ generated by admissible portfolios $\varphi(t); 0 \leq t \leq T$ in a market with the risky asset price process modeled as a semimartingale \item The optimal scenario $\frac{dQ^*}{dP}$ of the dual problem to minimize the expected $V$-value of $\frac{dQ}{dP}$ over a family of equivalent local martingale measures $Q$. Here $V$ is the convex dual function of the concave function $U$. \end{myenumerate} In this paper we consider markets modeled by Itô-Lévy processes, and in the first part we give a new proof of the above result in this setting, based on the maximum principle in stochastic control theory. An advantage with our approach is that it also gives an explicit relation between the optimal portfolio $\varphi^*$ and the optimal measure $Q^*$, in terms of backward stochastic differential equations. In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a relation between them. In particular, we show explicitly how to get from the solution of one of the problems to the solution of the other. We illustrate the results with explicit examples.