The cumulant representation of the Lundberg root in the case of semicontinuous processes

For the case of homogeneous processes $\xi (t)$, $\xi (0)=0$, $t\geq 0$, with independent increments and negative jumps, it is proved in A. V. Skorokhod, Random Processes with Independent Increments, Nauka, Moscow, 1964 that the functional \[ \tau ^+(x)=\inf \left \{t\geq 0\colon \xi (t)>x\right \}, \qquad x\geq 0, \] is a nondecreasing process with independent increments with respect to $x$, and its moment generating function is expressed via the cumulant that satisfies the corresponding Lundberg equation. The corresponding representations of this cumulant are specified and its Lévy characteristics (namely, $\gamma$ and Lévy’s integral measure $N(x)$) are evaluated by using some of the results of the author’s work of 2007 for the processes under consideration.