Abstract:
The first-exit time process of an inverse Gaussian Lévy process is
considered. The one-dimensional distribution functions of the process
are obtained. They are not infinitely divisible and the tail probabilities
decay exponentially. These distribution functions can also be viewed
as distribution functions of supremum of the Brownian motion with
drift. The density function is shown to solve a fractional PDE and
the result is also generalized to tempered stable subordinators. The
subordination of this process to the Brownian motion is considered
and the underlying PDE of the subordinated process is obtained. The
infinite divisibility of the first-exit time of a β-stable subordinator is
also discussed.
