Before revision: We study the generalized composite pure and randomized hypothesis testing problems. In addition to characterizing the corresponding optimal tests, we examine the conditions under which these two hypothesis testing problems are equivalent, and provide counterexamples when they are not. This analysis is useful for portfolio optimization problems that maximize some success probability given a fixed initial capital.
After revision: We study the portfolio problem of maximizing the outperformance probability over a random benchmark through dynamic trading with a fixed initial capital.