On matrix exponential approximations of the infimum of a spectrally negative Levy process

We recall four open problems concerning constructing high-order matrixexponential 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 CramérLundberg model, and recall a remarkable family of (not minimal order) approximations of Johnson and Taaffe [JT89], 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. keywords: Levy process; first passage problem; Pollaczek-Khinchine formula; method of moments; matrix-exponential function; admissible Padé approximation; Johnson-Taaffe approximations; two-point Padé approximations