Fluctuations of an omega-type killed process in discrete time

Abstract

The theory of the so-called ${\mathcal{W}_{q}}$ and ${\mathcal{Z}_{q}}$ scale functions is developped for the fluctuations of right-continuous discrete time and space killed random walks. Explicit expressions are derived for the resolvents and two-sided exit problem when killing depends on the present level of the process. Similar results in the reflected case are also considered. All the expressions are given in terms of new generalisations of the scale functions, which are obtained using arguments different from the continuous case (spectrally negative Lévy processes). Hence, the connections between the two cases are spelled out. For a specific form of the killing function, the probability of bankruptcy is obtained for the model known as omega model in the actuarial literature.

Keywords

• Fluctuation theory in discrete time
• exit problems
• upwards skip-free killed random walks
• potential measures
• one-step analysis
• probability of bankruptcy