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We discuss a multi-period portfolio selection problem with transaction costs in this paper. We assume that the sample space is finite, and the possible securities price vector transitions is equivalent to the number of securities. By introducing a set of auxiliary martingales, we connect the primal problem with a set of optimization problems without transaction costs. We find that the dual problem, which is to minimize the optimal value for the set of optimization problems, is equivalent to the primal problem, if it exists.