Two-dimensional Brownian risk model for cumulative Parisian ruin probability

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.