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This paper solves the general optimal prediction problem \\sup_{0\\leq\\tau\\leq T}\\sE[f(M_T-B_\\tau)], where the supremum is over all stopping times $\\tau$ adapted to the natural filtration of $(B_t)$, and $f$ is a nonincreasing convex function. The optimal stopping time $\\tau^*$ is shown to be of \"bang-bang\" type: $\\tau^*\\equiv 0$ if the drift of the underlying process $(B_t)$ is negative, and $\\tau^*\\equiv T$ is the drift is positive. 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Allaart","submitted_at":"2009-10-03T14:12:20Z","abstract_excerpt":"Let $(B_t)_{0\\leq t\\leq T}$ be either a Bernoulli random walk or a Brownian motion with drift, and let $M_t:=\\max\\{B_s: 0\\leq s\\leq t\\}$, $0\\leq t\\leq T$. This paper solves the general optimal prediction problem \\sup_{0\\leq\\tau\\leq T}\\sE[f(M_T-B_\\tau)], where the supremum is over all stopping times $\\tau$ adapted to the natural filtration of $(B_t)$, and $f$ is a nonincreasing convex function. The optimal stopping time $\\tau^*$ is shown to be of \"bang-bang\" type: $\\tau^*\\equiv 0$ if the drift of the underlying process $(B_t)$ is negative, and $\\tau^*\\equiv T$ is the drift is positive. 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