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Theory of Probability and Mathematical Statistics

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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
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 94 (2016).
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
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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

$\displaystyle W_{\gamma }(t) = X(t) - f(t) - \gamma \inf _{s\le t} (X(s) - f(s)),$

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.


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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

DOI: https://doi.org/10.1090/tpms/1017
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

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