  In this paper we present a duality theory for the robust utility maximisation
problem in continuous time for utility functions defined on the positive real
axis. Our results are inspired by -- and can be seen as the robust analogues of
-- the seminal work of Kramkov & Schachermayer [18]. Namely, we show that if
the set of attainable trading outcomes and the set of pricing measures satisfy
a bipolar relation, then the utility maximisation problem is in duality with a
conjugate problem. We further discuss the existence of optimal trading
strategies. In particular, our general results include the case of logarithmic
and power utility, and they apply to drift and volatility uncertainty.

Mathematical Finance;Optimization and Control;Probability