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

Abstract

This paper investigates $\pi_T(a_1,a_2) = \mathbb{P}\left(\sup\limits_{t\in[0,T]} (\sigma_1B(t)-c_1t)>a_1, \sup\limits_{t\in[0,T]}( \sigma_2 B(t)-c_2t)>a_2\right),$ where $\{B(t) : t \geq 0\}$ 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 $\pi_T\left(a_1,a_2\right)$ and find its asymptotic behavior both in the so called many-source and high-threshold regimes.