TY - JOUR
T1 - Omega diffusion risk model with surplus-dependent tax and capital injections
AU - Cui, Zhenyu
AU - Nguyen, Duy
N1 - Publisher Copyright:
© 2016 Elsevier B.V.
PY - 2016/5/1
Y1 - 2016/5/1
N2 - 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 (Albrecher et al., 2011), the risk model with tax (Albrecher and Hipp, 2007), and the risk model with capital injections (Albrecher and Ivanovs, 2014). 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 Li and Zhou (2013) and Zhang (2015). 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.
AB - 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 (Albrecher et al., 2011), the risk model with tax (Albrecher and Hipp, 2007), and the risk model with capital injections (Albrecher and Ivanovs, 2014). 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 Li and Zhou (2013) and Zhang (2015). 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.
KW - Azéma-Yor process
KW - Occupation time
KW - Omega risk model
KW - Reflected diffusions
KW - Risk model with tax
KW - Time-homogeneous diffusion
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U2 - 10.1016/j.insmatheco.2016.03.012
DO - 10.1016/j.insmatheco.2016.03.012
M3 - Article
AN - SCOPUS:84962361049
SN - 0167-6687
VL - 68
SP - 150
EP - 161
JO - Insurance: Mathematics and Economics
JF - Insurance: Mathematics and Economics
ER -