A generalized Neyman-Pearson lemma for g-probabilities

This paper is concerned with hypothesis tests for g-probabilities, a class of nonlinear probability measures. The problem is shown to be a special case of a general stochastic optimization problem where the objective is to choose the terminal state of certain backward stochastic differential equations so as to minimize a g-expectation. The latter is solved with a stochastic maximum principle approach. Neyman-Pearson type results are thereby derived for the original problem with both simple and randomized tests. It turns out that the likelihood ratio in the optimal tests is nothing else than the ratio of the adjoint processes associated with the maximum principle. Concrete examples, ranging from the classical simple tests, financial market modelling with ambiguity, to super- and sub-pricing of contingent claims and to risk measures, are presented to illustrate the applications of the results obtained. © 2009 Springer-Verlag.