  We study a generalized risk process $X(t)=Y(t)-C(t)$, $t\in[0,\tau]$, where
$Y$ is a L\'evy process, $C$ an independent subordinator and $\tau$ an
independent exponential time. Dropping the standard assumptions on the finite
expectations of the processes $Y$ and $C$ and the net profit condition, we
derive a Pollaczek-Khinchine type formula for the supremum of the dual process
$\widehat{X}=-X$ on $[0,\tau]$ which generalizes the results obtained in
\cite{HPSV1}. We also discuss which assumptions are necessary for deriving this
formula, specially from the point of view of the ladder process.
