Finite-time ruin probability for correlated Brownian motions

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 $\rho \in (-1,1)$ and define the joint survival probability of both supremum functionals $\pi_\rho(c_1,c_2; u, v)$ by $$\pi_\rho(c_1,c_2; u, v)=\mathbb{P}\left(\sup_{s \in [0,1]} \left(W_1(s)-c_1s\right)>u,\sup_{t \in [0,1]} \left(W_2(t)-c_2t\right)>v\right) ,$$ where $c_1,c_2 \in \mathbb{R}$ and $u,v$ are given positive constants. Approximation of $\pi_\rho(c_1,c_2; u, v) $ is of interest for the analysis of ruin probability in bivariate Brownian risk model as well as in the study of bivariate test statistics. In this contribution we derive tight bounds for $\pi_\rho(c_1,c_2; u, v)$ in the case $\rho \in (0,1)$ and obtain precise approximations by letting $u\to \infty$ and taking $v= au$ for some fixed positive constant $a$ and $\rho \in (-1,1).$