Finite-time ruin probability for correlated Brownian motions

Let $(W_1(s), W_2(t)), s,t\ge 0 $(W1(s),W2(t)),s,t≥0 be a two-dimensional Gaussian process with standard Brownian motion marginals and constant correlation $\rho \in (-1,1) $ρ∈(−1,1). Define the joint survival probability of both supremum functionals by \[ \pi_\rho(c_1,c_2; u, v)=\pk{\sup_{s \in [0,1]} \left(W_1(s)-c_1s\right) \gt u,\sup_{t \in [0,1]} \left(W_2(t)-c_2t\right) \gt v}, \]πρ(c1,c2;u,v)=Psups∈[0,1]W1(s)−c1s>u,supt∈[0,1]W2(t)−c2t>v, where $c_1,c_2 \in \mathbb {R} $c1,c2∈R and u, v are given positive constants. Approximation of $\pi _\rho (c_1,c_2; u, v) $πρ(c1,c2;u,v) is of interest for the analysis of ruin probability in bivariate Brownian risk model, as well as in the study of the power of bivariate test statistics. In this contribution, we derive tight bounds for $\pi _\rho (c_1,c_2; u, v) $πρ(c1,c2;u,v) in the case $\rho \in (0,1) $ρ∈(0,1) and obtain precise approximations for all $\rho \in (-1,1) $ρ∈(−1,1) by letting $u\to \infty $u→∞ and taking v = au for some fixed positive constant a.