# Distribution of suprema for generalized risk processes

## TL;DR

This paper derives a generalized Pollaczek-Khinchine formula for the distribution of the supremum of a risk process involving Lévy and subordinator components, relaxing standard assumptions and analyzing ladder process conditions.

## Contribution

It introduces a new formula for the supremum distribution of a generalized risk process without requiring finite expectations or net profit conditions.

## Key findings

- Derived a Pollaczek-Khinchine type formula for the supremum distribution.
- Identified necessary assumptions related to ladder processes.
- Extended previous results to more general risk process models.

## Abstract

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.