Under convenient geometric assumptions, the saddle-point method for
multidimensional Laplace integrals is extended to the case where the
contours of integration have boundaries. The asymptotics are studied
in the case of nondegenerate and of degenerate isolated critical
points. The incidence of the Stokes phenomenon is related to the
monodromy of the homology via generalized Picard-Lefschetz formulae
and is quantified in terms of geometric indices of intersection. Exact
remainder terms and the hyperasymptotics are then derived. A direct
consequence is a numerical algorithm to determine the Stokes constants
and indices of intersections. Examples are provided.