  We consider a discrete-time robust utility maximisation with semistatic
strategies, and the associated indifference prices of exotic options. For this
purpose, we introduce a robust form of convex integral functionals on the space
of bounded continuous functions on a Polish space, and establish some key
regularity and representation results, in the spirit of the classical
Rockafellar theorem, in terms of the duality formed with the space of Borel
measures. These results (together with the standard Fenchel duality and minimax
theorems) yield a duality for the robust utility maximisation problem as well
as a representation of associated indifference prices, where the presence of
static positions in the primal problem appears in the dual problem as a
marginal constraint on the martingale measures. Consequently, the resulting
indifference prices are consistent with the observed prices of vanilla options.
