Generalized inverse Gaussian distributions and the time of first level crossing

We propose a new approximation for the distribution of the time of the first crossing of a high level $u$ by random process $\text{homV}s-cs$, where $\text{homV}s$, $s>0$, is compound renewal process and $c>0$. It significantly outperforms the existing approximations, particularly in the region around the critical point $c=\text{cS}$ which separates processes with positive and negative drifts. This approximation is tightly related to generalized inverse Gaussian distributions.