Exact asymptotics of component-wise extrema of two-dimensional Brownian motion

Abstract

We derive the exact asymptotics of \[ P\left( \sup_{t\ge 0} \Bigl( X_1(t) - \mu_1 t\Bigr)> u, \ \sup_{s\ge 0} \Bigl( X_2(s) - \mu_2 s\Bigr)> u \right), \ \ u\to\infty, \] where $(X_1(t),X_2(s))_{t,s\ge0}$ is a correlated two-dimensional Brownian motion with correlation $\rho\in[-1,1]$ and $\mu_1,\mu_2>0$. It appears that the play between $\rho$ and $\mu_1,\mu_2$ leads to several types of asymptotics. Although the exponent in the asymptotics as a function of $\rho$ is continuous, one can observe different types of prefactor functions depending on the range of $\rho$, which constitute a phase-type transition phenomena.