  In this paper we give sufficient conditions guaranteeing the validity of the
well-known minimax theorem for the lower Snell envelope with respect to a
family of absolutely continuous probability measures. Such minimax results play
an important role in the characterisation of arbitrage-free prices of American
contingent claims in incomplete markets. Our conditions do not rely on the
notions of stability under pasting or time-consistency and reveal some
unexpected connection between the minimax result and the path properties of the
corresponding density process.
