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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 43, Fasc. 1,
pages 63 - 81
DOI: 10.37190/0208-4147.00090
Published online 2.6.2023
 

Cumulative Parisian ruin probability for two-dimensional Brownian risk model

K. Krystecki

Abstract:

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 $u \to \IF$ for the cumulative Parisian ruin probability \pk∗∫[0,1]1(W1(s)−c1s>u)ds>H1(u)\vspace∗2pt∫[0,1]1(W2(t)−c2t>au)dt>H2(u) for c1, c2 ∈ ℝ, a ∈ (0, 1] and suitably adjusted H1(u), H2(u).

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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 $u \to \IF$ for the cumulative Parisian ruin probability \pk∗∫[0,1]1(W1(s)−c1s>u)ds>H1(u)\vspace∗2pt∫[0,1]1(W2(t)−c2t>au)dt>H2(u) for c1, c2 ∈ ℝ, a ∈ (0, 1] and suitably adjusted H1(u), H2(u).

2010 AMS Mathematics Subject Classification: Primary 60G15; Secondary 60G70.

Keywords and phrases: multidimensional Brownian motion, stationary random fields, extremes.

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