Properties of American option prices

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On Threshold Strategies and the Smooth-Fit Principle for Optimal Stopping Problems (2007, background): “A general result comes from these papers: the value function is convex if either • δ is constant, that is, (e−(r−δ)tXt )t≥0 is a local martingale [9], [18], [11], [7], or • xδ(x) is nondecreasing and e−rtXt converges to 0 in L1 (see [1, Corollary 1]).”
Bubbles, convexity and the Black–Scholes equation (2009, background): “Indeed, if g is concave, then e−rtg(X(t)) is a supermartingale, thus implying that V (x, t) = g(x) (see [6] for the case when α satisfies (6)).”