Parisian ruin over a finite-time horizon

Abstract

For a risk process R u (t) = u + ct − X(t), t ≥ 0, where u ≥ 0 is the initial capital, c > 0 is the premium rate and X(t), t ≥ 0 is an aggregate claim process, we investigate the probability of the Parisianwith a given positive constant S and a positive measurable function T u . We derive asymptotic expansion of P S (u, T u ), as u → ∞, for the aggregate claim process X modeled by Gaussian processes. As a by-product, we derive the exact tail asymptotics of the infimum of a standard Brownian motion with drift over a finite-time interval.

Approximation of ruin probability and ruin time in discrete Brownian risk models

“…For the continuous time [13] gives an exact formula for the Parisian ruin probability. Both finite and infinite Parisian ruin times for continuous setup of the problem are dealt with in [14,15].…”

“…Section 3 discusses the ruin probability for the γ-reflected Brownian risk model, see also [9][10][11][12]. The approximation of Parisian ruin (see [13][14][15]) and cumulative Parisian ruin (see [16][17][18])…”

Uniform tail approximation of homogenous functionals of Gaussian fields

“…determines the asymptotics of the Parisian ruin of the corresponding risk model; see [13]. Note that the classical Piterbarg constant corresponds to the S = 0 case.…”

“…This uniform counterpart of (1.2) is crucial when the processes X u,τ u are parameterized by u and τ u . see [11] and [13], or inf s∈A u sup t∈B u Y (s, t), see [7] and [8].…”

Extremes of threshold-dependent Gaussian processes

“…are the Pickands and Piterbarg constants, respectively, where B α is a standard fractional Brownian motion (fBm) with self-similarity index α/2 ∈ (0, 1], see [19][20][21][22][23][24][25] for properties of both constants.…”