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Generalized Hypothesis Testing and Maximizing the Success Probability in Financial Markets

We study the generalized composite pure and randomized hypothesis testing problems. In addition to characterizing the 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 to maximize some success probability given a ﬁxed initial capital. 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.