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

We derive the exact asymptotics of $$P\left(\underset{t\ge 0}{\sup }\right(X_1(t)-{\mu }_{1}t)>u,\underset{s\ge 0}{\sup }(X_2(s)-{\mu }_{2}s)>u),u\to \mathrm{\infty },$$ where $(X_1(t),X_2(s){)}_{t,s\ge 0}$ 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.