UNIVERSITY
OF WROC£AW
 
Main Page
Contents
Online First
General Information
Instructions for authors


VOLUMES
43.2 43.1 42.2 42.1 41.2 41.1 40.2
40.1 39.2 39.1 38.2 38.1 37.2 37.1
36.2 36.1 35.2 35.1 34.2 34.1 33.2
33.1 32.2 32.1 31.2 31.1 30.2 30.1
29.2 29.1 28.2 28.1 27.2 27.1 26.2
26.1 25.2 25.1 24.2 24.1 23.2 23.1
22.2 22.1 21.2 21.1 20.2 20.1 19.2
19.1 18.2 18.1 17.2 17.1 16.2 16.1
15 14.2 14.1 13.2 13.1 12.2 12.1
11.2 11.1 10.2 10.1 9.2 9.1 8
7.2 7.1 6.2 6.1 5.2 5.1 4.2
4.1 3.2 3.1 2.2 2.1 1.2 1.1
 
 
WROC£AW 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 β‰₯ 0, be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation Οβ€„βˆˆβ€„(βˆ’1,1). We derive the exact asymptotics as $u \to \IF$ for the cumulative Parisian ruin probability $$\pk*{\begin{array}{@{\,}l@{\,}}\int_{[0,1]} \mathbf{1}(W_1(s)-c_1s>u)\,ds>H_1(u) \vspace*{2pt}\\ \int_{[0,1]} \mathbf{1}(W_2(t)-c_2t>au)\,dt>H_2(u)\end{array}}$$ for c1, c2β€„βˆˆβ€„β„, aβ€„βˆˆβ€„(0, 1] and suitably adjusted H1(u), H2(u).

============================

Let (W1(s),W2(t)), s, t β‰₯ 0, be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation Οβ€„βˆˆβ€„(βˆ’1,1). We derive the exact asymptotics as $u \to \IF$ for the cumulative Parisian ruin probability $$\pk*{\begin{array}{@{\,}l@{\,}}\int_{[0,1]} \mathbf{1}(W_1(s)-c_1s>u)\,ds>H_1(u) \vspace*{2pt}\\ \int_{[0,1]} \mathbf{1}(W_2(t)-c_2t>au)\,dt>H_2(u)\end{array}}$$ for c1, c2β€„βˆˆβ€„β„, aβ€„βˆˆβ€„(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.

Download:        Full text