  In this paper, we provide duality bounds for the maxitive integral of a
non-additive set function. These bounds allow for a monotone analogue of the
basic results in large deviations theory. In particular, we show the
equivalence of the large deviation principle and the Laplace principle for
general concentration functions and characterize the respective monotone rate
function. As an application of the results, we obtain on the one hand a
monotone version of Cram\'{e}r's theorem, and on the other hand a large
deviation and Laplace principle for the asymptotic comparison of random walks
as introduced by Fritz [8].
