On the infimum attained by the reflected fractional Brownian motion

Abstract

Let {B H (t) : t ≥ 0} be a fractional Brownian motion with Hurst parameter H ∈ 1 2 , 1 . For the storage processas u → ∞. This finding, known in the literature as the strong Piterbarg property, goes in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.

Parisian ruin of self-similar Gaussian risk processes

“…Proof of Lemma 6.1: Note that the sup-inf functional satisfies F1-F2 in [13]. The proof follows by similar arguments as the proof of Lemma 1 therein, and therefore we omit the technical details.…”

“…We first present a crucial lemma which can be seen as an extension of the celebrated Pickands lemma; see, e.g., [34,35,36]. We refer to [13] for recent developments in this direction.…”

Extremes of stationary Gaussian storage models

“…As a result, we extend findings of Dieker (2005), where the asymptotics of P (Q(0) > u) was considered. Moreover we generalize Piterbarg (2001) and Dębicki and Kosiński (2014) where the exact asymptotics of ψ sup T u (u) and ψ inf T u (u) were studied for fractional Brownian motion model with β = 1. As a by-product we find conditions under which the strong Piterbarg property phenomena Eq.…”

“…as u → ∞, providing that H > 1/2 and T u = o(u 2H −1 H ); see Piterbarg (2001) and Dębicki and Kosiński (2014). Property (2) is nowadays referred to as the strong Piterbarg property.…”

Piterbarg theorems for chi-processes with trend

“…We refer to Dȩbicki (2002) and Dȩbicki and Kosiński (2014) for other generalizations of the Pickands or Piterbarg constants. Organization of the paper: The main results for the stationary and non-stationary chi-processes with trend are given in Section 2.…”