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 $${\pi }_{\rho }(c_1,c_2;u,v)=\mathbb{P}\left(\underset{s\in [0,1]}{\sup }\left(W_1(s)-c_{1}s\right)>u,\underset{t\in [0,1]}{\sup }\left(W_2(t)-c_{2}t\right)>v\right),$$ 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 $\rho \in (-1,1).$

35 pages