  The main goal of this article is to construct "arithmetic Okounkov bodies"
for an arbitrary pseudo-effective (1,1)-class $\alpha$ on a K\"ahler manifold.
Firstly, using Boucksom's divisorial Zariski decompositions for
pseudo-effective (1,1)-classes on compact K\"ahler manifolds, we prove the
differentiability of volumes of big classes for K\"ahler manifolds on which
modified nef cones and nef cones coincide; this includes K\"ahler surfaces. We
then apply our differentiability results to prove Demailly's transcendental
Morse inequality for these particular classes of K\"ahler manifolds. In the
second part, we construct the convex body $\Delta(\alpha)$ for any big class
$\alpha$ with respect to a fixed flag by using positive currents, and prove
that this newly defined convex body coincides with the Okounkov body when
$\alpha\in {\rm NS}\_{\mathbb{R}}(X)$; such convex sets $\Delta(\alpha)$ will
be called generalized Okounkov bodies. As an application we prove that any
rational point in the interior of Okounkov bodies is "valuative". Next we give
a complete characterisation of generalized Okounkov bodies on surfaces, and
show that the generalized Okounkov bodies behave very similarly to original
Okounkov bodies. By the differentiability formula, we can relate the standard
Euclidean volume of $\Delta(\alpha)$ in $\mathbb{R}^2$ to the volume of a big
class $\alpha$, as defined by Boucksom; this solves a problem raised by
Lazarsfeld in the case of surfaces. Finally, we study the behavior of the
generalized Okounkov bodies on the boundary of the big cone, which are
characterized by numerical dimension.
