Two-dimensional Brownian risk model for cumulative Parisian ruin probability

TL;DR

This paper derives precise approximations for the probability of cumulative Parisian ruin in a two-dimensional Brownian risk model, conditioned on the occurrence of ruin, focusing on asymptotic behavior related to the process's time spent over barriers.

Contribution

It provides new asymptotic approximations for the probability of cumulative Parisian ruin in a correlated two-dimensional Brownian motion model, conditioned on ruin occurrence.

Findings

Derived explicit asymptotic formulas for ruin probabilities.

Analyzed the impact of correlation and barrier functions on ruin probabilities.

Provided insights into the time spent over barriers before ruin occurs.

Abstract

Let $(W_1(s), W_2(t)), s,t\ge 0$ be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation $\rho \in (-1,1).$ In this contribution we derive precise approximations for cumulative Parisian ruin conditioned on the occurrence of the ruin of the aforementioned two-dimensional Brownian motion, i.e. $$\mathbb{P}\left(\begin{array}{ccc}\int_{[0,1]} \mathbf{1}(W_1^*(s)>u)ds>H_1(u) \\ \int_{[0,1]} \mathbf{1}(W_2^*(t)>au)dt>H_2(u)\end{array}\Bigg{|}\exists_{v,w \in [0,1]}\begin{array}{ccc} W_1(v)-c_1v>u \\ W_2(w)-c_2w>au \end{array}\right).$$ We study the asymptotics for specific functions $\boldsymbol{H}(u)$ for $u$ being proportional to initial position of the Brownian motion, which determines how long does the process need to spend over the barrier.