The distribution of the supremum of a $\gamma$-reflected stochastic process with an input process belonging to some exponential type Orlicz space

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.