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Theory of Probability and Mathematical Statistics
(Teoriya Imovirnostei ta Matematychna Statystyka)



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

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