Theory of Probability and Mathematical Statistics
Inequalities for a risk function in the time and space inhomogeneous Cramer-Lundberg risk model
M. V. Kartashov, V. V. Golomoziy
Download PDF
Abstract: We consider the generalization of the Cramer-Lundberg risk model with the Poisson process of risk events which is inhomogeneous in time, and with current reserve dependent premium rate. We obtain explicit inequalities for the exponential normed uniform distances between risk functions in (a) time-space inhomogeous model and space-inhomogeneous time-homogeneous model, (b) space-inhomogeneous time-homogeneous model and space and time-homogeneous model. It is supposed some closeness of the premium rate to a constant.
Keywords: inhomogeneous in time and space Markov process, Cramer-Lundberg risk model, ruin probability.
Bibliography: 1. E. B. Dynkin, Markov processes, v.1 and v.2, Academic Press, New York; Springer, Berlin,1966.
2. I. I. Gikhman, A. V. Skorokhod, The theory of stochastic processes II, Springer-Verlag Berlin Heidelberg, 1983.
3. V. M. Zolotarev, Modern theory of sums of independent random variables, Nauka, Moscow, 1986.
4. V. M. Shurenkov, Ergodic Markov processes, Nauka, Moscow, 1989. (Russian)
5. J. Grandell, Aspects of risk theory, Springer Series in Statistics, 1991.
6. L. N. Bowers, H. U. Gerber, J. C. Hickman, D. A. Jones, C. J. Nesbitt, Actuarial Mathematics, The Society of Actuaries, 1997.
7. S.P. Mayn, R. L. Tweedie, Markov chains and stochastic stability, Springer-Verlag, 1993.
8. N. V. Kartashov, Strong stable Markov chains, VSP, Utrecht, 1996.
9. N. V. Kartashov, The stability of almost homogeneous in time Markov semigroups of operators, Theory Probab. Math. Statist., 71 (2004), 119-128.
10. N. V. Kartashov, The ergodicity and stability of quasi-homogeneous Markov semigroups of operators, Theory Probab. Math. Statist., 72 (2006), 59-68.
11. N. V. Kartashov, O. M. Stroev, Lundberg approximation for the risk function in an almost homogeneous environment, Theory Probab. Math. Statist., 73 (2006), 71-79.
12. N. V. Kartashov, The stability of transient quasi-homogeneous Markov semigroups and an estimate of the ruin probability, Theory Probab. Math. Statist., 75 (2007), 41-50.
13. N. V. Kartashov, Boundness, limits, and stability of solutions of a perturbation of a nonhomogeneous renewal equation on a semiaxis, Theory Probab. Math. Statist., 81 (2010), 71-83.
