  For a multivariate L\'evy process satisfying the Cram\'er moment condition
and having a drift vector with at least one negative component, we derive the
exact asymptotics of the probability of ever hitting the positive orthant that
is being translated to infinity along a fixed vector with positive components.
This problem is motivated by the multivariate ruin problem introduced in F.
Avram et al. (2008) in the two-dimensional case. Our solution relies on the
analysis from Y. Pan and K. Borovkov (2017) for multivariate random walks and
an appropriate time discretization.
