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We study the convex duality method for robust utility maximization in thepresence of a random endowment. When the underlying price process is a locallybounded semimartingale, we show that the fundamental duality relation holdstrue for a wide class of utility functions on the whole real line and unboundedrandom endowment. To obtain this duality, we prove a robust version ofRockafellar's theorem on convex integral functionals and apply Fenchel'sgeneral duality theorem.

Duality in Robust Utility Maximization with Unbounded Claim via a Robust Extension of Rockafellar's Theorem