On the gamma-reflected processes with fBm input

Define a $γ$-reflected process $W_γ(t)=Y_H(t)-γ\inf_{s\in[0,t]}Y_H(s)$, $t\ge0$ with input process $\{Y_H(t), t\ge 0\}$ which is a fractional Brownian motion with Hurst index $H\in (0,1)$ and a negative linear trend. In risk theory $R_γ(t)=u-W_γ(t), t\ge0$ is referred to as the risk process with tax of a loss-carry-forward type, whereas in queueing theory $W_1$ is referred to as the queue length process. In this paper, we investigate the ruin probability and the ruin time of the risk process $R_γ, γ\in [0,1]$ over a surplus dependent time interval $[0, T_u]$.

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