Theory of Probability and Mathematical Statistics
The distribution of the supremum of a gamma-reflected stochastic process with an input process belonging to some exponential type Orlicz space
R. Ye. Yamnenko
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Abstract: The paper is devoted to investigation of properties of a $\gamma$-reflected process with input process from a certain Orlicz space of exponential type, namely sub-Gaussian, $\varphi$-sub-Gaussian and random input processes from general classes $V(\varphi,\psi)$ are studied. The $\gamma$-reflected process is a random process of type $W_{\gamma}(t) = X(t) - f(t) - \gamma \inf_{s\le t} (X(s) - f(s))$, where $f(t)$ is some function. This process arises in insurance mathematics as a risk process for which income taxing is conducted via loss-carry-forward scheme by paying a proportion $\gamma \in [0,1]$ of incoming premiums when the process is on its maximum. The case of $\gamma < 0$ can be considered as a model with stimulation proportional to the increase of maximum and a value $\gamma > 1$ can be interpreted as corresponding model with inhibition.Ruin probability estimates $\P\left\{\sup_{t}W_{\gamma}(t) >x \right\}$ of the corresponding risk model are studied for all $\gamma\in\R$. Obtained results are applied to a sub-Gaussian generalized fractional Brownian motion process.
Keywords: Generalized fractional Brownian motion, metric entropy, distribution estimate, sub-Gaussian process, Orlicz space
Bibliography: 1.H. Albreher, J. Ivanovs, Power identities for Levy risk models under taxation and capital injections, Stochastic Systems. Vol. 4, No. 1 (2014), 157-172.
2.V. Buldygin, Convergence of random elements in topological spaces, "Naukova Dumka", Kiev, 1983.
3.V. Buldygin, Yu. Kozachenko, Sub-Gaussian random vectors and processes, Theory Probab. Math. Stat. 36 (1988), 9-20.
4.V. Buldygin and Yu. Kozachenko, Metric Characterization of Random Variables and Random Processes. AMS, Providence, RI, 2000.
5.K. Debicki, M. Mandjes, Exact overflow asymptotics for queues with many Gaussian inputs. Journal of Applied Probability, 40 (2003), 704-720.
6.E. Hashorva, L. Ji, V. Piterbarg, On the Supremum of gamma-reflected Processes with Fractional Brownian Motion as Input, Stochastic Processes and their Applications, 123(11) (2014), 4111-4127.
7.J.P. Kahane, Proprietes locales des fonctions a series de Fouries aleatories, Studia Math. 19, No. 1 (1960), 1-25.
8.Yu. Kozachenko, E. Ostrovskii, Banach spaces random variables of subGaussian type, Probability Theory and Math. Statistics, 32 (1986), 45-56.
9.Yu. Kozachenko, O. Vasylyk, Random processes from classes $V(\varphi,\psi)$. Probability Theory and Math. Statistics, 63 (2000), 100-111.
10.Yu. Kozachenko, O. Vasylyk and R. Yamnenko, Upper estimate of overrunning by $Sub_\varphi(\Omega)$ random process the level specified by continuous function, Random Oper. and Stoch. Equ., 13 (2005), No. 2, 111-128.
11.Vasylyk, O.I., Kozachenko, Yu.V., Yamnenko, R.E., $\varphi$-sub-Gaussian random processes. Kyiv: Vydavnycho-Poligrafichnyi Tsentr, Kyivskyi Universytet, 2008.
12.Yu. Kozachenko, R. Yamnenko, Application of $\varphi$-sub-Gaussian random processes in Queuing Theory, in: Modern trends in Stochastics, Springer Optimization and Its Applications, 90, Springer, Berlin (2014), 21-38.
13.V.I. Piterbarg. Asymptotic methods in the theory of Gaussian processes and fields. In: Transl. Math. Monographs, vol. 148 (1996). AMS, Providence, RI.
14.R. Yamnenko, Bounds for the distribution of some functionals of processes with $\varphi$-sub-Gaussian increments, Theory Probab. Math. Stat., 85 (2012), 181-197.
15.R. Yamnenko, On distribution of the norm of deviation of a sub-Gaussian random process in Orlicz spaces, Random Operators and Stochastic Equations, Vol. 23, Iss. 3 (2015), 187-194.
16.R. Yamnenko and O. Vasylyk, Random process from the class $V(\varphi,\psi)$: exceeding a curve, Theory of Stochastic Processes, 13 (29) (2007), No. 4, 219-232.
17. R. Yamnenko and O. Shramko, On the distribution of storage processes from the class V($\varphi$, $\psi$), Theory Probab. Math. Stat., 83 (2011), 191-206.
18.R. Yamnenko, Yu. Kozachenko and D. Bushmitch, Generalized sub-Gaussian fractional Brownian motion queueing model, Queueing Systems, 77(1) (2014), 75-96.
