We obtain closed-form expressions for the prices and optimal hedging strategies of American put-options in the presence of an ``up-and-out" barrier , both with and without constraints on the short-selling of stock. The constrained case leads to a stochastic optimization problem of mixed optimal stopping/singular control type. This is reduced to a variational inequality which is then solved explicitly in two qualitatively separate cases, according to a certain compatibility condition among the market coefficients and the constraint.
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