The distribution of the size of the first jump over a level for a class of stochastic processes

The paper deals with a stochastic process with independent increments with the characteristic function
$$ \varphi(t,z)=\exp\biggl\{t\biggl[i\alpha z-\frac{\sigma^2}{2}z^2+\lambda\int_0^{\infty}(e^{izx}-1)F\,(dx)\biggr]\biggr\}. $$
For the distribution of the size of the first jump over a level $x>0$, a) an integro-differential equation (in $x$) is obtained, b) the limiting behaviour is studied as $x\to\infty$ and c) the Laplace transform (in $x$) is found.