On the γ-reflected processes with fBm input*

Abstract

which is a fractional Brownian motion with Hurst index H ∈ (0, 1) and a negative linear trend. In risk theory R γ (t) = u − W γ (t), t ≥ 0 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 over a surplus dependent time interval [0, T u ].

An Erdös–Révész type law of the iterated logarithm for reflected fractional Brownian motion

“…In this paper we analyze 0-1 properties of a class of such processes, that due to its importance in queueing theory (and dual risk theory) gained substantial interest; see, e.g., Norros (2004), Piterbarg (2001), Asmussen (2003), and Asmussen and Albrecher (2010) or novel works on γ -reflected Gaussian processes (Hashorva et al 2013;Liu et al 2015).…”

Extremes of stationary Gaussian storage models

“…Motivated by the above applications, Q(0) has been studied in the literature under different levels of generality, e.g., Norros (1994), Hüsler and Piterbarg (1999), Dębicki (2002), Hüsler and Piterbarg (2004), Dieker (2005), Hashorva et al (2013), and Liu et al (2015). Particularly vast interest has been paid to the analysis of storage models, where X(t) = B H (t) is a fractional Brownian motion (fBm) with Hurst index H ∈ (0, 1) and β = 1, leading to derivation of exact asymptotics of P (Q(0) > u) as u → ∞ in Hüsler and Piterbarg (1999) and a surprising asymptotic equivalence P sup…”

Extremes ofγ-reflected Gaussian processes with stationary increments

“…If ϕ = 0, then lim u→∞ ∆ γ (u) = 0 for γ ∈ (0, 1], implying that only the local behaviour of σ 2 at 0 contributes to the limit in (22). If ϕ ∈ (0, ∞), then lim u→∞ ∆ γ (u) ∈ (0, ∞), indicating that the whole function σ 2 determines the limit in (22). If ϕ = ∞, then lim u→∞ ∆ γ (u) = ∞, which means that the value of σ 2 (t) as t → ∞ is sufficient for the limit in (22).…”

“…If ϕ ∈ (0, ∞), then lim u→∞ ∆ γ (u) ∈ (0, ∞), indicating that the whole function σ 2 determines the limit in (22). If ϕ = ∞, then lim u→∞ ∆ γ (u) = ∞, which means that the value of σ 2 (t) as t → ∞ is sufficient for the limit in (22).…”

“…The asymptotics in (5) shows that the generalised Piterbarg constant governs the relation between the two ruin probabilities corresponding to the model with tax and without tax, i.e., it defines what we call the asymptotic tax equivalence. However, in view of [21,22] we know that for the case X = B H , the tax rate γ does not influence the limiting distribution of the first and the last passage times. We investigate these problems in more general models for X.…”