  We recall four open problems concerning constructing high-order
matrix-exponential approximations for the infimum of a spectrally negative Levy
process (with applications to first-passage/ruin probabilities, the waiting
time distribution in the M/G/1 queue, pricing of barrier options, etc). On the
way, we provide a new approximation, for the perturbed Cramer-Lundberg model,
and recall a remarkable family of (not minimal order) approximations of Johnson
and Taaffe, which fit an arbitrarily high number of moments, greatly
generalizing the currently used approximations of Renyi, De Vylder and
Whitt-Ramsay. Obtaining such approximations which fit the Laplace transform at
infinity as well would be quite useful.
