The distribution of the supremum of a $\gamma$-reflected stochastic process with an input process belonging to some exponential type Orlicz space
Author:
R. E. Yamnenko
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 94 (2017), 185-201
MSC (2010):
Primary 60G07; Secondary 60K25
DOI:
https://doi.org/10.1090/tpms/1017
Published electronically:
August 25, 2017
MathSciNet review:
3553462
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Abstract:
The paper is devoted to the study of properties of a $\gamma$-reflected process with an input belonging to some exponential type Orlicz space. In particular, sub-Gaussian and $\varphi$-sub-Gaussian whose input processes belong to some of the general classes $V(\varphi ,\psi )$ are studied. The $\gamma$-reflected process is a stochastic process of the form \[ W_{\gamma }(t) = X(t) - f(t) - \gamma \inf _{s\le t} (X(s) - f(s)),\] where $f(t)$ is a given function. This kind of process arises in insurance mathematics as a model for risk processes for which the income tax is paid according to the loss-carry-forward scheme where a proportion $\gamma \in [0,1]$ of incoming premiums is paid when the process is on its maximum. The case of $\gamma < 0$ corresponds to a model with stimulation proportional to the increase of maximum, while the case of $\gamma > 1$ can be interpreted as a corresponding model with inhibition.
Some upper bounds for the ruin probability $\mathsf {P}\left \{\sup _{t}W_{\gamma }(t) >x \right \}$ are considered in the corresponding risk model for all $\gamma \in \mathbb {R}$. The results obtained in the paper are applied for the case of the sub-Gaussian generalized fractional Brownian motion.
References
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References
- H. Albreher and J. Ivanovs, Power identities for Lévy risk models under taxation and capital injections, Stoch. Systems 4 (2014), no. 1, 157–172. MR 3353216
- V. V. Buldygin, Convergence of Random Elements in Topological Spaces, “Naukova Dumka”, Kiev, 1980. (Russian) MR 734899
- V. V. Buldygin and Yu. V. Kozachenko, Sub-Gaussian random vectors and processes, Teor. Veroyatnost. i Mat. Statist. 36 (1987), 10–22, 138; English transl. in Theory Probab. Math. Stat. 36 (1988), 9–20. MR 913713
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, “TBiMS”, Kiev, 1998; English transl. American Mathematical Society, Providence, RI, 2000. MR 1743716
- K. Debicki and M. Mandjes, Exact overflow asymptotics for queues with many Gaussian inputs, J. Appl. Probab. 40 (2003), 704–720. MR 1993262
- E. Hashorva, L. Ji, and V. Piterbarg, On the supremum of gamma-reflected processes with fractional Brownian motion as input, Stoch. Process. Appl. 123(11) (2014), 4111–4127. MR 3091101
- J. P. Kahane, Propriétés locales des fonctions à series de Fouries aléatories, Studia Math. 19, (1960), no. 1, 1–25. MR 0117506
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- Yu. Kozachenko, O. Vasylyk, and R. Yamnenko, Upper estimate of overrunning by $\operatorname {Sub}_\varphi (\Omega )$ random process the level specified by continuous function, Random Oper. Stochastic Equations 13 (2005), no. 2, 111–128. MR 2152102
- Yu. Kozachenko, O. Vasylyk, and R. Yamnenko, $\varphi$-sub-Gaussian Stochastic processes, “Kyiv University”, Kyiv, 2008. (Ukrainian)
- Yu. Kozachenko and R. Yamnenko, Application of $\varphi$-sub-Gaussian random processes in queuing theory, Modern Trends in Stochastics, Springer Optimization and Its Applications vol. 90, Springer, Berlin, 2014, pp. 21–38. MR 3236066
- V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, Transl. Math. Monographs, vol. 148, American Mathematical Society, Providence, RI, 1996. MR 1361884
- R. Yamnenko, Bounds for the distribution of some functionals of processes with $\varphi$-sub-Gaussian increments, Teor. Ĭmovir. Mat. Stat. 85 (2011), 161–176; English transl. in Theory Probab. Math. Stat. 85 (2012), 181–197. MR 2933713
- R. Yamnenko, On distribution of the norm of deviation of a sub-Gaussian random process in Orlicz spaces, Random Oper. Stochastic Equations 23 (2015), no. 3, 187–194. MR 3393431
- R. Yamnenko and O. Vasylyk, Random process from the class $V(\varphi ,\psi )$: exceeding a curve, Theory Stoch. Process. 13 (29) (2007), no. 4, 219–232. MR 2482262
- R. Yamnenko and O. Shramko, On the distribution of storage processes from the class V($\varphi , \psi$), Teor. Ĭmovir. Mat. Stat. 83 (2010), 163–176; English transl. in Theory Probab. Math. Stat. 83 (2011), 191–206. MR 2768858
- R. Yamnenko, Yu. Kozachenko, and D. Bushmitch, Generalized sub-Gaussian fractional Brownian motion queueing model, Queueing Systems 77(1) (2014), 75–96. MR 3183511
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Additional Information
R. E. Yamnenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
yamnenko@univ.kiev.ua
Keywords:
Generalized fractional Brownian motion,
metric entropy,
an estimate of the distribution,
sub-Gaussian process,
Orlicz space
Received by editor(s):
March 21, 2016
Published electronically:
August 25, 2017
Article copyright:
© Copyright 2017
American Mathematical Society