An Upper Bound of Large Deviations for Capacities

Up to now, most of the academic researches about the large deviation and risk theory are under the framework of the classical linear expectations. But motivated by problems of model uncertainties in statistics, measures of risk, and superhedging in finance, sublinear expectations are extensively studied. In this paper, we obtain a type of large deviation principle under the sublinear expectation. This result is a new expression of the Gärtner-Ellis theorem under the sublinear expectations which is in the classical theory of large deviations. In addition, we introduce a new process under the sublinear expectations, that is, the -Poisson process. We give an application of our result and obtain the rate function of the compound -Poisson process in the upper bound of large deviations for capacities. The application of our result opens a new field for the research of risk theory in the future.