We study the discrete time risk process modelled by the skip-free random walk and derive results connected to the ruin probability and crossing a fixed level for this type of process. We use the method relying on the classical ballot theorems to derive the results for crossing a fixed level and compare them to the results known for the continuous time version of the risk process. We generalize this model by adding a perturbation and, still relying on the skip-free structure of that process, we generalize the previous results on crossing the fixed level for the generalized discrete time risk process. We further derive the famous Pollaczek-Khinchine type formula for this generalized process, using the decomposition of the supremum of the dual process at some special instants of time.
Addario-Berry, L. and Reed, B. A., Ballot theorems, old and new, Horizons of Combinatorics, Bolyai Society Mathematical Studies17 (2008), 9–35.
Addario-Berry, L. and Reed, B. A., Ballot theorems, old and new, Horizons of Combinatorics, Bolyai Society Mathematical Studies17 (2008), 9–35.)| false
Lambert, A., Some aspects of discrete branching processes, Université Paris VI Pierre et Marie Curie, France (2010) http://archive.schools.cimpa.info/archivesecoles/20101207164945/coursalambert.pdf)| false