Running supremum of Brownian motion in dimension 2: exact and asymptotic results

Abstract

This note investigates ${\pi }_{T,c_1,c_2}(a_1,a_2)=\mathbb{P}(\underset{t\in [0,T]}{\sup }({\sigma }_{1}B(t)-c_{1}t)>a_1,\underset{t\in [0,T]}{\sup }({\sigma }_{2}B(t)-c_{2}t)>a_2),$ where $\{B(t):t\ge 0\}$ is a standard Brownian motion, with $T>0,{\sigma }_1,{\sigma }_2>0,c_1,c_2\in \mathbb{R}.$ We derive explicit formula for the probability ${\pi }_{T,c_1,c_2}(a_1,a_2)$ and find its asymptotic behavior both in the so called many-source and high-threshold regimes.