The exercise payoff of an American capped call with the cap L is given by max(min(S, L) − X, 0), L > X. Let S^∗_cap(τ) and S^∗(τ) denote the early exercise boundary of the American capped call and its noncapped counterpart, respectively. Show that (Broadie and Detemple, 1995)

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Scap (t) = min (S* (T), L).