Some inequalities for the risk function in the time and space nonhomogeneous Cramér–Lundberg risk model

We consider a generalized Cramér–Lundberg risk model with a time nonhomogeneous Poisson process of insurance claims where the intensity of premiums depends on the current insurance capital. Some explicit bounds are found for the exponential normalized uniform distance between

[(a)] the risk function in a time nonhomogeneous model and that in a time homogeneous but space nonhomogeneous model,

[(b)] the risk functions in a time homogeneous model or in a space nonhomogeneous model and that in a time and space homogeneous model.

It is assumed that the intensity of premiums approaches a constant as the current insurance capital increases.

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Additional Information

M. V. Kartashov
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email: nkartashov@skif.com.ua

V. V. Golomozyĭ
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email: vitaliy.golomoziy@gmail.com

Keywords: Time and space nonhomogeneous Markov processes, ruin probability, Cramér–Lundberg risk model
Received by editor(s): November 2, 2017
Published electronically: August 19, 2019
Additional Notes: M. V. Kartashov passed away on January 1, 2018. See the tribute to him by the Theory of Probability and Mathematical Statistics Editorial Board in this issue
Article copyright: © Copyright 2019 American Mathematical Society