Robust utility maximization in terms of supermartingale measures

We consider a dual description of the optimal value of robust utility in the abstract model of the financial market $(\Omega,\mathscr{F},\mathrm{P},\mathscr{A}(x))$, where $\mathscr{A}(x)=x\mathscr{A}$, $x\geq 0$, is the set of the investor's terminal capitals corresponding to strategies with the initial capital $x$. The main result of the paper addresses the question of the transition in the definition of the dual problem from the polar of the set $\mathscr{A}$ to a narrower set of limit values of supermartingale densities.