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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).

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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.

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