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Cumulative Parisian ruin probability for two-dimensional Brownian risk model

Krystecki Konrad

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 1

63--81

Let (W1(s), W2(t)), s, t ≤ 0, be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation ρ ∈ (−1, 1). We derive the exact asymptotics as u → ∞ for the cumulative Parisian ruin probability [Formula] for c1, c2 ∈ R, a ∈ (0, 1] and suitably adjusted H1(u), H2(u).

Pickands–Piterbarg Constants for Self-Similar Gaussian Processes

Dębicki Krzysztof, Tabiś Kamil

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 2

297--315

For a centered self-similar Gaussian process {Y (t) : t ∈ [0;∞)} and R≥0 we analyze the asymptotic behavior of [formula], for suitably chosen γ> 0. Additionally, we find bounds for HRY , R > 0, and a surprising relation between HY and the classical Pickands constants.