  We study the scale function of the spectrally negative phase-type Levy
process. Its scale function admits an analytical expression and so do a number
of its fluctuation identities. Motivated by the fact that the class of
phase-type distributions is dense in the class of all positive-valued
distributions, we propose a new approach to approximating the scale function
and the associated fluctuation identities for a general spectrally negative
Levy process. Numerical examples are provided to illustrate the effectiveness
of the approximation method.
