  S. G. Kou and H. Wang [First Passage times of a Jump Diffusion Process
\textit{Ann. Appl. Probab.} {\bf 35} (2003) 504--531] give expressions of both
the (real) Laplace transform of the distribution of first passage time and the
(real) Laplace transform of the joint distribution of the first passage time
and the running maxima of a jump-diffusion model called Kou model. %However, to
invert the last Laplace transform it is needed Kuo and Wang invert the first
Laplace transform by using Gaver-Stehfest algorithm, and the inversion of
second one involves a large computing time with an algebra computer system. In
the present paper, we give a much simpler expression of the Laplace transform
of the joint distribution, and we also show, using Complex Analysis techniques,
that both Laplace transform can be extended to the complex plane. Hence, we can
use variants of the Fourier-series methods to invert that Laplace transfoms,
which are very efficent. The improvement in the computing times and accuracy is
remarkable.
