  We study the portfolio problem of maximizing the outperformance probability
over a random benchmark through dynamic trading with a fixed initial capital.
Under a general incomplete market framework, this stochastic control problem
can be formulated as a composite 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.

Portfolio Management;Applications