On the supremum ofγ-reflected processes with fractional Brownian motion as input

Abstract

Let {X H (t), t ≥ 0} be a fractional Brownian motion with Hurst index H ∈ (0, 1] and define a γ-In this paper we establish the exact tail asymptotic behaviour of M γ (T ) = sup t∈[0,T ] W γ (t) for any T ∈ (0, ∞].Furthermore, we derive the exact tail asymptotic behaviour of the supremum of certain non-homogeneous mean-zero Gaussian random fields.

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).…”

Approximation of Passage Times of γ-Reflected Processes with FBM Input

“…With motivation from the aforementioned contributions, this paper is concerned with the Gaussian approximation of the random vector (τ * 1 (u), τ * 2 (u)). For the derivation of the tail asymptotics of sup t∈[0,T ] W γ (t), Hashorva et al (2013) showed that the investigation of the supremum of certain nonstationary Gaussian random fields is crucial. One key merit of our problem of approximating the joint distribution function of (τ * 1 (u), τ * 2 (u)) is that it leads, as in the case of the analysis of the tail asymptotics of sup t∈[0,T ] W γ (t), to an interesting unsolved problem of asymptotic theory of Gaussian random fields.…”

On the infimum attained by the reflected fractional Brownian motion

“…The shape of Lemma 1 is tailored to the needs of the next section, where asymptotics of tail distribution of inf sup functionals of Gaussian processes are analyzed. Various further extensions of Lemma 1 can be thought of along the lines of already existing extensions of the classical Pickands' lemma, especially in the direction allowing nonconstant variance function of the family (X u ), as in Piterbarg and Prisyazhnyuk (1978) or Hashorva et al (2013).…”

“…In this paper we analyze the asymptotic properties of tail distribution of infimum of an important class of such processes, that naturally appear in models of storage (queueing) systems and, by duality to ruin problems, gained broad interest also in problems arising in finance and insurance risk; see, e.g., Norros (2004), Piterbarg (2001), Asmussen (2003), and Asmussen and Albrecher (2010) or a novel work (Hashorva et al 2013).…”