Before revision:  The corresponding dual is related to a pure hypothesis testing problem which may or may not coincide with the randomized hypothesis testing problem . Our framework is applicable to both complete and incomplete market settings .
After revision:  The corresponding dual is related to a pure hypothesis testing problem . We analyze the connection between this pure testing problem and its randomized counterpart, and from latter we derive a dual representation for the maximal outperformance probability. Moreover, in a complete market setting, we provide a closed-form solution to the problem of beating a leveraged exchange traded fund. For a general benchmark under an incomplete stochastic factor model, we provide the Hamilton-Jacobi-Bellman PDE characterization for the maximal outperformance probability .