Omega diffusion risk model with surplus-dependent tax and capital injections. (English) Zbl 1369.91080
Summary: In this paper, we propose and study an Omega risk model with a constant bankruptcy function, surplus-dependent tax payments and capital injections in a time-homogeneous diffusion setting. The surplus value process is both refracted (paying tax) at its running maximum and reflected (injecting capital) at a lower constant boundary. The new model incorporates practical features from the Omega risk model [H. Albrecher et al., Eur. Actuar. J. 1, No. 1, 43–55 (2011; Zbl 1219.91062)], the risk model with tax [H. Albrecher and C. Hipp, Bl. DGVFM 28, No. 1, 13–28 (2007; Zbl 1119.62103)], and the risk model with capital injections [H. Albrecher and J. Ivanovs, Stoch. Syst. 4, No. 1, 157–172 (2014; Zbl 1300.60067)]. The study of this new risk model is closely related to the Azéma-Yor process, which is a process refracted by its running maximum. We explicitly characterize the Laplace transform of the occupation time of an Azéma-Yor process below a constant level until the first passage time of another Azéma-Yor process or until an independent exponential time. We also consider the case when the process has a lower reflecting boundary. This result unifies and extends recent results of B. Li and X. Zhou [Adv. Appl. Probab. 45, No. 4, 1049–1067 (2013; Zbl 1370.60136)] and H. Zhang [Adv. Appl. Probab. 47, No. 1, 210–230 (2015; Zbl 1310.60114)]. We explicitly characterize the Laplace transform of the time of bankruptcy in the Omega risk model with tax and capital injections up to eigen-functions, and determine the expected present value of tax payments until default. We also discuss a further extension to occupation functionals through stochastic time-change, which handles the case of a non-constant bankruptcy function. Finally we present examples using a Brownian motion with drift, and discuss the pricing of quantile options written on the Azéma-Yor process.
MSC:
| 91B30 | Risk theory, insurance (MSC2010) |
| 60G51 | Processes with independent increments; Lévy processes |
Keywords:
time-homogeneous diffusion; Azéma-Yor process; occupation time; risk model with tax; omega risk model; reflected diffusionsReferences:
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