Fluctuation theory of omega-killed Lévy processes

Date

2024-3

Author

Şimşek, Meral

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This thesis explores the fluctuation identities for omega-killed (reflected) processes. In the first part, we develop the theory of the so-called $\mathcal{W}_q$ and $\mathcal{Z}_q$ scale functions for the fluctuations of right-continuous discrete time and space killed random walks. Explicit expressions are derived for the resolvents and two-sided exit problems when killing depends on the present level of the process. Similar results in the reflected case are also considered. All the derivations are given in terms of a new generalization of the scale functions, which are obtained using different arguments from the continuous case (spectrally negative L\'evy processes). Hence, we spell out the connections between the two cases. We obtain the probability of bankruptcy in the omega model of the actuarial literature for a specific form of the killing function. In the second part of this thesis, we analyze exit problems for a level-dependent L\'evy process, which is exponentially killed with an intensity depending on the present state of the process. By considering level-dependent spectrally negative Lévy processes, we take general premium rate as a function depending on the current level of the processes as well. Further, we derive the respective resolvents for the omega-killed level-dependent process. All exit identities are introduced via a new generalization of scale functions, $\mathpzc{W}^{(q)}$ and $\mathpzc{Z}^{(q)}$ (counterparts of the scale function from the theory of Lévy processes), which are solutions of Volterra integral equations. Additionally, we obtain similar results for the reflected omega-killed level-dependent Lévy processes. The existence of the solution of the stochastic differential equation for reflected level-dependent Lévy processes is also discussed. Finally, to illustrate our result, the probability of bankruptcy is obtained for an insurance risk process.

Subject Keywords

Fluctuation theory, Omega-killed lévy process, Upwards skip-free random walk, Multi-refracted process, Level-dependent process

URI

https://hdl.handle.net/11511/109407

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Graduate School of Natural and Applied Sciences, Thesis